Math 18 Precalculus Mathematics Module 1 PDF

UNIVERSITY OF THE PHILIPPINES VISAYAS College of Arts and Sciences Division of Physical Sciences and Mathematics MATH 18: Precalculus Mathematics First Semester, A.Y. 2024-2025...

UNIVERSITY OF THE PHILIPPINES VISAYAS College of Arts and Sciences Division of Physical Sciences and Mathematics MATH 18: Precalculus Mathematics First Semester, A.Y. 2024-2025 MODULE 1 The Real Number System, Properties of the Set of Real Numbers; Inequalities and Intervals; Quadratic, Rational, and Absolute Value Inequalities; Two-Dimensional Coordinate System, Slope and Distance; Forms of Equations of Lines; Relations and Functions, Types of Functions; Operations on Functions, Inverse of a Function 1 Real Number System Definition 1.0.1. The Set of Natural/Counting Numbers The set of counting numbers or also referred to as natural numbers which we denote by N, may be written N = {1, 2, 3,...} The set of natural numbers is also called as the set of positive integers and is denoted as Z+. Definition 1.0.2. The Set of Whole Numbers The elements of the set of natural numbers N, together with the number 0, form the set of whole numbers and is denoted by W W = {0} ∪ N = {0, 1, 2, 3,...} Definition 1.0.3. The Set of Negative Integers For each element x of Z+ , there corresponds a number −x which is called a negative integer. We denote the set of negative integers by Z− Z− = {... , −3, −2, −1} Definition 1.0.4. The Set of Integers The set of integers, denoted by Z, is a set whose elements are positive integers (Z+ ), negative integers (Z− ) and the element 0. Z = Z− ∪ {0} ∪ Z+ = {... , −3, −2, −1, 0, 1, 2, 3,...} Remarks. The number 0, which is an integer, is neither positive nor negative. Math 18 | Page 1  Definition 1.0.5. The Set of Rational Numbers A Rational Number is a number that can be expressed as a ratio of two integers, where the denominator is non-zero. The set of rational numbers, denoted by Q is defined as   p Q = x x = ; p, q ∈ Z; q ̸= 0 q Remarks. The following are rational numbers: 1. All integers are rational but not all rational numbers are integers. 0 Number 0, an integer, can be expressed as 0 = where n ̸= 0, n ∈ Z. n n Any integer n, can be expressed as a ratio of itself and an integer 1, i.e., n =. 1 2. Terminating decimals 25 71234 0.25 = 7.1234 = 100 10000 1414 15463789 1.414 = 154.63789 = 1000 100000 3. Non-terminating but repeating decimals 122 10 421 1.2323... = 0.62588 = 99 16 650 Definition 1.0.6. The Set of Irrational Numbers A number which cannot be expressed as a ratio between two integers is called an irrational number. The set of irrational numbers is denoted by Q′. Remark: Decimals which are Non-terminating and Non-repeating are classified to be irrational. Example. 1. π = 3.14159... 3. 2.11211121111211111... √ 2. 2 = 1.41421... 4. e = 2.71828... Definition 1.0.7. The Set of Real Numbers The union of the set of rational and irrational number is the set of real numbers. We denote the set of real numbers by R and write R = Q ∪ Q′ Math 18 | Page 2  Real Numbers R = Q ∪ Q′ Rational Irrational Numbers Numbers Q′ Q Integers Other Fractions Z = {... , −2, −1, 0, 1, 2,... } (non-integers) Whole Numbers Negative Integers W = {0, 1, 2,... } Z− Natural Numbers N = {1, 2, 3,... } Zero {0} 2 Properties of Real Numbers 2.1 Equality Axioms Definition 2.1.1. Equality Axioms For any a, b, c ∈ R, the following Equality Axioms holds 1. Reflexivity for Equality: a = a. 2. Symmetry for Equality: if a = b then b = a. 3. Transitivity for Equality: if a = b and b = c then a = c. 4. Addition Property for Equality: if a = b then a + c = b + c. 5. Multiplication Property for Equality: if a = b then a · c = b · c. 2.2 Field Axioms Definition 2.2.1. Real Number System The real number system consists of the set of real number and two operations, addition (+) and multiplication (·). Math 18 | Page 3  Definition 2.2.2. Field Axioms For any a, b, c ∈ R, the following field axioms hold: 1. Closure Addition: The sum of two real numbers a and b, denoted by a + b, is also a real number. Multiplication: The product of two real numbers a and b, denoted by a · b, is also a real number. 2. Associativity Addition: (a + b) + c = a + (b + c) Multiplication: (a · b) · c = a · (b · c) 3. Commutativity Addition: a + b = b + a Multiplication: a · b = b · a 4. Distributivity of Multiplication over Addition Left Hand: c · (a + b) = (c · a) + (c · b) Right Hand: (a + b) · c = (a · c) + (b · c) 5. Existence of Identity Elements Additive Identity: There exists a unique number 0 such that a + 0 = a. Multiplicative Identity: There exists a unique number 1 such that a · 1 = a. 6. Existence of Inverses Additive Inverse: There exists a unique number −a (read as “negative of a”) for any real number a such that a + (−a) = 0. 1 Multiplicative Inverse: There exists a unique number for any real number a 1 a ̸= 0 such that a · = 1. a Example. Let x, y, z ∈ R. Give the axiom which justifies each of the following statements. 1. (a + 3)x = ax + 3x 2. 5 + (−5) = 0 3. If x + y = z and z = 10 then x + y = 10. 4. 8 + y = y + 8 5. (4x) · y = 4 · (xy) Definition 2.2.3. Subtraction Subtraction is the operation that undoes the addition; to subtract a number from another, we simply add the negative of that number. By definition a − b = a + (−b) Math 18 | Page 4  Definition 2.2.4. Division Division is the operation that undoes the multiplication; to divide by a number, we multiply by the multiplicative inverse of that number, i.e., if b ̸= 0 then a 1 =a· b b 2.3 Properties of Negatives Definition 2.3.1. Properties of Negatives For any a, b ∈ R, the following properties for negatives hold: 1. −(−a) = a 2. (−a)b = a(−b) = −(ab) 3. (−1)a = −a 4. (−a)(−b) = ab 5. −(a + b) = −a − b 6. −(a − b) = b − a Example. Prove the properties 1 to 6 for negatives. 2.4 Properties of Fractions Definition 2.4.1. Properties of Fractions For any a, b, c, d ∈ R, the following properties of fractions hold: a 1 1. =1 6. =a a 1 a 1 1 1 2. · = a a d a b ab 7. b = · c a c ac d b c 3. · = b d bd a b a+b ac a 8. + = 4. = c c c bc b a c a c ad + bc 5. If = then ad = bc 9. + = b d b d bd Example. 1. Prove that for any real numbers a, b, c, if a + c = b + c, then a = b. 2. Prove that for any real numbers a, b with a ̸= 0, if ab = a, then b = 1. 2.5 Order Axioms Real Numbers posses ordering relation which is illustrated by the symbol “>”, which is read as greater than. Math 18 | Page 5  Definition 2.5.1. Order Axioms For any a, b, c ∈ R, the following Order Axioms holds 1. Trichotomy Axiom: One and only one of the following statements hold true a > b, b > a or a = b. 2. Transitivity for Inequality: If a > b and b > c then a > c. 3. Addition Rule for Inequalities: If a > b then a + c > b + c. 4. Multiplication Rule for Inequalities: If a > b and c > 0 then ac > bc. Definition 2.5.2. Positive or Negative Numbers A real number is said to be Positive if a > 0 and negative if 0 > a. Definition 2.5.3. Less than For any real numbers a, b, we say that a < b, read as “a is Less than b” if and only if b > a. Remarks. For any real number a, 1. a is positive if and only if a > 0. 2. a is negative if and only if a < 0. 2.6 Basic Theorems on Order Definition 2.6.1. Basic Theorems on Order For any a, b, c, d ∈ R, the following Basic Theorem on Order holds 1. Closure of Positive Numbers: Addition: If a > 0 and b > 0 then a + b > 0. Multiplication: If a > 0 and b > 0 then ab > 0. 2. If a is a positive real number then −a is negative, and if a is negative then −a is positive. 3. If a > b then −b > −a. 4. If a > b and c < 0 then bc > ac. 1 5. If a > 0 then > 0. a 6. If a < b and c < d then a + c < b + d. 1 1 7. Reciprocal Rule: If a < b and both are positive (or negative) then >. a b 2.7 One-Dimensional Coordinate System Definition 2.7.1. One-Dimensional Coordinate System The One-Dimensional Coordinate System is a geometric interpretation of real numbers. Each real number is associated with a point along a horizontal line to form the real number line. Math 18 | Page 6 Remarks. 1. A one-to-one correspondence between the set of real numbers and the set of all points in a line can be established. Locating/ Plotting the Points: Start with the counting numbers Natural N (zero may be included). 0 1 2 3 Extend the line backward to include the negatives. Integers Z −3 −2 −1 0 1 2 3 Insert all the fractions. Rational Q −3− 11 −2 − 4 −1 − 1 0 1 1 4 2 11 3 4 3 2 2 3 4 Insert all the roots. √ 2 −3− 11 −2 − 4 −1 − 1 0 1 1 4 2 11 3 4 3 2 2 3 4 Fill in all the numbers to make a √ continuous line Real 2 e π R −3− 11 −2 − 4 −1 − 1 0 1 1 4 2 11 3 4 3 2 2 3 4 2. Numbers to the right of zero are all positive and negative to the left. 3. Numbers to the right of a certain number are always larger. 4. Numbers to the left of a certain number are always smaller. Definition 2.7.2. Interval Notation 1. Open Interval: (a, b) = {x ∈ R|a < x < b} a b 2. Closed Interval: [a, b] = {x ∈ R|a ≤ x ≤ b} a b 3. Half Open (Half Closed) Interval: [a, b) = {x ∈ R|a ≤ x < b} a b (a, b] = {x ∈ R|a < x ≤ b} a b Math 18 | Page 7  4. Open Ray Interval: (a, +∞) = {x ∈ R|x > a} a (−∞, b) = {x ∈ R|x < b} b 5. Closed Ray Interval: [a, +∞) = {x ∈ R|x ≥ a} a (−∞, b] = {x ∈ R|x ≤ b} b Example. Graph each set. 1. (2, 4) ∩ [3, 8] 2. (2, 4) ∪ [3, 8] 3 Inequalities and Intervals; Quadratic, Rational, and Ab- solute Value Inequalities 3.1 Inequalities and the Interval Notation Definition 3.1.1. Inequality An inequality is a statement saying that one expression is less than or equal to another. Definition 3.1.2. Inequality in One Variable An inequality in one variable is a statement involving two expressions at least one contain- ing the variable separated by one of the inequality symbols , ≥. Example. 1. x + 1 > 3 x−3 4. ≥4 2 2. 4 − 2x < 3(3 − x) 2x − 7 3. x2 − 2x ≥ 3 5. ≥3 x−5 3.2 Solutions of Inequalities Definition 3.2.1. Domain The set of real numbers for which the members if the inequality is defined is the domain. Math 18 | Page 8  Definition 3.2.2. Solutions of Inequality These admissible values of the variable, if any that result in a true statement are called solutions of inequality. These are the value or values that would make the inequality a true statements. Remarks. 1. To solve an inequality means to find all the solutions of the inequality by the use of algebraic manipulations. 2. The idea is to convert or transform the given equation in an equivalent inequalities obtained by algebraic manipulations, which are inequalities having the same solution sets. Definition 3.2.3. Absolute Inequality Absolute inequality is an inequality that is true for all permissible values, that is all values in the domain is part of the solution set. Example. 1. x < x + 1 2. |x| ≥ x 3. x < |x| + 1 4. x2 ≥ x Definition 3.2.4. Conditional Inequality Conditional inequality is an inequality that is true only for some values, that is there exists at leas one element of the domain which is not part of the solution set. Example. 1. 2 > x 2. x + 1 > 0 Recall: For any a, b, c ∈ R, 1. Addition Rule for Inequalities: If a > b then a + c > b + c. 2. Multiplication Rule for Inequalities: If a > b and c > 0 then ac > bc. 3. If a > b and c < 0 then bc > ac. 3.3 Linear Inequality Definition 3.3.1. Linear Inequality An inequality is linear if each term is constant or a multiple of the variable. To solve a linear inequality, isolate the variable on one side of the inequality. Example. 1. 4 − 2x < 3(3 − x) 2. −1 ≤ 3x + 5 ≤ 2 Math 18 | Page 9 3.4 Polynomial and Rational Inequality Definition 3.4.1. Solving Polynomial and Rational Inequality 1. The inequality should be transformed first to standard form, that is the right side is zero and the left side is in the factored form. 2. By considering the equality of the expression instead on inequality the Critical Numbers of the inequality can be solved. 3. Perform sign analysis. Definition 3.4.2. Critical Numbers Critical numbers are numbers that can either make the entire expression be equal to zero or make the expressions undefined. Remarks. 1. Always put is standard form, that is all expressions should be equated to zero. 2. Never multiply both sides by any expression containing the unknown variable. Example. Find the solution set of the following inequalities. 1. x2 − 2x ≥ 3 2. x2 − 2x ≥ −1 2x2 − 7x − 4 3. ≤0 x−1 3.5 Absolute Value Inequality Definition 3.5.1. Absolute Value The absolute value of a number a, denoted by |a|, is the distance from a to 0 on the real number line. Distance is always positive or zero, so we have |a| ≥ 0 for every number a. If a is a real number, then the Absolute Value of a is  a a≥0 |a| = −a a < 0 Definition 3.5.2. Properties of Absolute Value For any a, b ∈ R, the following properties hold: 1. |a| ≥ 0 a |a| 4. = , b ̸= 0 b |b| 2. |a| = | − a| 5. |a + b| ≤ |a| + |b| 3. |ab| = |a||b| 6. |a| − |b| ≤ |a − b| Math 18 | Page 10  Definition 3.5.3. Properties of Inequalities Involving Absolute Value For any a ∈ R, the following properties hold: 1. |x| < a if and only if −a < x < a or alternatively, x < a and x > −a. 2. |x| ≤ a if and only if −a ≤ x ≤ a or alternatively, x ≤ a and x ≥ −a. 3. |x| > a if and only if x > a or x < −a. 4. |x| ≥ a if and only if x ≥ a or x ≤ −a. Example. Find the solutions of the following absolute inequalities. |x − 3| 5 − 3x 1. >4 2. ≤7 2 x+2 4 Two Dimensional Coordinate System, Slope, and Distance 4.1 Two Dimensional Coordinate System The two-dimensional coordinate system consists 3 y of points on the plane determined by the two per- pendicular coordinate axis. 2 (x, y) ➤ The horizontal line is the x-axis while the 1 vertical line is the y-axis. ➤ The point of intersection of these coordinate (0, 0) x axes is called the origin. −3 −2 −1 1 2 3 Each point is uniquely associated to an ordered −1 pair in the set R × R = {(a, b)|a, b ∈ R} and vice versa. −2 Thus, there is one-to-one correspondence between −3 the set R × R and the set of points on the plane. Each pair associated uniquely to a point on the plane is called the coordinates of the point. ➤ The first component of the point is called the abscissa or the x-coordinate, and the second component is called the ordinate or the y-coordinate. The origin corresponds to the ordered pair (0, 0). Elements of the set R × R having the form (a, 0) corresponds to the points lying on the x-axis. ➤ If a > 0, then the points lie to the right of the origin while if a < 0, the points lie to the left of the origin. Elements of the set R × R having the form (0, b) corresponds to the points lying on the y-axis. ➤ If b > 0, then the points lie above the origin while if b < 0, the points lie below the origin. Math 18 | Page 11  The plane is divided into four quadrants. y 3 Quadrant II Quadrant I 2 (−, +) (+, +) 1 (0, 0) - Origin x −5 −4 −3 −2 −1 1 2 3 4 5 −1 Quadrant III Quadrant IV −2 (−, −) (+, −) −3 Example. Determine the quadrant or axis where the following points can be found. 1. (−5, −3) 3. (1, −11) 5. (−4, 0) 2. (0, 5) 4. (2, 7) 4.2 Distance, Midpoint, and Slope Formulas Let P1 (x1 , y1 ) and P2 (x2 , y2 ) be points on the Cartesian plane. Definition 4.2.1. Distance Formula The distance between P1 and P2 is given by p P1 P2 = P2 P1 = (x2 − x1 )2 + (y2 − y1 )2. Definition 4.2.2. Midpoint Formula The midpoint of P1 P2 is given by   x1 + x2 y1 + y2 ,. 2 2 Definition 4.2.3. Slope Formula The slope of P1 P2 is given by y2 − y1 y1 − y2 mP1 P2 = = , x2 − x 1 x1 − x 2 provided that x1 ̸= x2. Remarks. The slope of a vertical line is undefined. The slope of a horizontal line is 0. Parallel lines have equal slopes. Math 18 | Page 12  Slopes of perpendicular lines are negative reciprocal of each other. Or, the product of the slopes of perpendicular lines is −1. 3 y 3 y 2 2 1 1 x x −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 −1 −1 −2 −2 −3 −3 Example. 1. Which of the points A(2, −1) or B(5, 6) is closer to the point C(4, 2)? 2. What is the midpoint of the line segment joining (3, 5) and (−1, 3)? 3. What is the slope of the line segment joining the points A(3, −1) and B(−1, 2)? 4.3 Applications of Distance, Slope, and Midpoint Definition 4.3.1. 1. Problem: Show that the given line segments are parallel. Hint: If AB ∥ CD, then mAB = mCD. 2. Problem: Show that the given line segments are perpendicular. Hint: If AB ⊥ CD, then mAB · mCD = −1. 3. Problem: Find the perimeter of the polygon given the coordinates of its vertices. Hint: To get the perimeter, construct the polygon and get the length of each side by using the distance formula. 4. Problem: Show that given points are collinear. Hint 1: By distance formula: Points A, B, and C are collinear if AB + BC = AC , given that point B is between A and C. Hint 2: Slope formula: Points A, B, and C are collinear if mAB = mBC = mAC. 5. Problem: Show that a triangle is isosceles given its vertices. Hint: Show that two sides are equal by the distance formula. 6. Problem: Show that a triangle is a right triangle given its vertices. Hint 1: Distance formula: Show that the sides satisfy the Pythagorean Theorem. Hint 2: Slope formula: Show that two sides are perpendicular. Math 18 | Page 13  7. Problem: Show that the given quadrilateral is a parallelogram. Hint 1: Distance formula: Show that the opposite sides are equal. Hint 2: Slope formula: Show that the opposite sides are parallel. Hint 3: Midpoint formula: Show that the diagonals have a common midpoint; hence they bisect each other at that point. 8. Problem: Show that the parallelogram is a rectangle. Hint 1: Distance formula: Show that diagonals are equal. Hint 2: Slope formula: Show that adjacent sides are perpendicular. 9. Problem: Show that the rectangle is a square. Hint 1: Distance formula: Show that all sides are equal. Hint 2: Slope formula: Show that diagonals are perpendicular. Example. 1. Show that the points A(1, −2), B(0, 1), and C(−1, 4) are collinear. 2. Show that the given triangle is a right triangle, given its vertices are A(−3, 2), B(−3, −2), and C(−7, −2). 3. Let AB ∥ CD. Given A(−2, 2), B(4, 8), C(−1, −1), and D(x, 2), find x. 5 Forms of Equations of Lines 5.1 Line Definition 5.1.1. A line is the set of all points that satisfy a first degree equation in R2. y Definition 5.1.2. x-intercept The x-coordinates of the points where a graph intersects the x-axis are called the x-intercepts of the graph and are obtained by setting y = 0 in the equation of the graph. b (0, b) (a, 0) x Definition 5.1.3. y-intercept a The y-coordinates of the points where a graph intersects the y-axis are called the y-intercepts of the graph and are obtained by setting x = 0 in the equation of the graph. Example. Find the x- and y-intercepts of the graph of the equation y = 3x − 4. Math 18 | Page 14 5.2 Forms of Equations of Lines Definition 5.2.1. Forms of Equations of Lines A. General Form of the Equation of a Line The general form of the equation of a line is Ax + By + C = 0 where A, B, and C are real numbers, A and B are not both zero. B. Slope-Intercept Form of the Equation of a Line An equation of the line that has slope m and y-intercept b is y = mx + b. C. Point-Slope Form of the Equation of a Line An equation of the line that passes through the point (x1 , y1 ) and has slope m is y − y1 = m(x − x1 ). D. Two-Point Form of the Equation of a Line An equation of the line that passes through the points (x1 , y1 ) and (x2 , y2 )   y2 − y1 y − y1 = (x − x1 ). x2 − x1 E. Intercept Form of the Equation of a Line An equation of the line that has x-intercept a and y-intercept b is x y + = 1, a b where a, b ̸= 0. Example. 1 1. Find the general equation of the line having x-intercept and y-intercept −3. 2 2. Find the slope and y-intercept of the line 3y + 2x = 6. 3. Find the general equation of the line that passes through the points (3, −2) and (1, 6). Recall: Let m1 and m2 be the slopes of the nonvertical lines L1 and L2. Then L1 and L2 are 1 parallel if and only if m1 = m2 , while L1 and L2 are perpendicular if and only if m2 = −. m1 Example. Let L be the line 4x + 3y − 6 = 0. a. Find the general equation of the line L1 parallel to L through the point P (−1, 4). b. Find the general equation of the line L2 perpendicular to L through the point Q(2, −3). Math 18 | Page 15 6 Relations Definition 6.0.1. Relation X Y A relation R from X to Y , denoted by R : X → 1 a Y , is a set of ordered pairs (x, y) such that for 2 each x ∈ X, there corresponds at least one y ∈ Y. 3 b Example. Consider the sets A = {1, 2, 3} and B = {3, 4, 5}. Determine which among the following is a relation from A to B. 1. X = {(4, 1), (5, 2), (3, 3)} 2. Y = {(1, 3), (2, 4), (3, 5)} 3. Z = {(1, 3), (2, 3), (3, 3)} Definition 6.0.2. Domain of a Relation The domain of the relation R, denoted by domR, is the set composed of the first components of all the ordered pairs in R. It is also defined as the set of all values for the independent variable in R. In symbols, domR = {x ∈ X|∃y ∈ Y with (x, y) ∈ R}. Definition 6.0.3. Range of a Relation The range of the relation R, denoted by ranR, is the set composed of the second components of all the ordered pairs in R. It is also defined as the set of all values for the dependent variable in R. In symbols, ranR = {y ∈ Y |∃x ∈ X with (x, y) ∈ R}. Definition 6.0.4. Ways of Representing a Relation Set of ordered pairs: e.g. {(3, 4), (1, 2), (5, 2), (3, 1)}, {(1, 4), (1, 2), (2, 2), (2, 1)} Equations/Inequalities: √ e.g. y = 5x + 1, y = 2x − 3, |y| + |x| = 7, y > x Graph: e.g. y y x x Math 18 | Page 16 Definition 6.0.5. Kinds of Relations 1. One-to-One X Y For each element in Y , there is one corresponding element of X. 2. One-to-Many X Y For one element of X, there is more than one corresponding element of Y. 3. Many-to-One X Y For an element in Y , there is more than one element of X. 4. Many-to-Many X Y For an element of X, there is more than one corresponding element of Y and vice versa. Math 18 | Page 17 7 Functions Definition 7.0.1. Function A function F from X to Y , denoted by F : X → Y , is a set of ordered pairs (x, y) such that for each x ∈ X, there corresponds a unique y ∈ Y. Remarks. A function is a special relation which stresses that no two distinct ordered pairs should have the same first component. Not all relations are functions, but all functions are relations. Example. Going back to the kinds of relations, which are functions? Remarks. 1. Given a set of ordered pairs, the first coordinate should not be repeated for the entire set to be a function. 2. Given an equation, it is a function if: y is present y has no even exponent y has no absolute value symbol 3. Inequalities are mere relations. Definition 7.0.2. Vertical Line Test The vertical line test is a test to determine if a relation is a function. A relation is a function if there are no vertical lines that intersect the graph at more than one point. Example. Which of the following relations are functions? Ordered Pairs: 1. {(3, 4), (1, 2), (5, 5), (3, 1)} 4. {(1, 2), (2, 3), (3, 4), (4, 5)} 2. {(1, 4), (1, 2), (2, 2), (2, 1)} 5. {(1, 1), (2, 2), (3, 3), (2, 2)} 3. {(1, 2), (2, 2), (3, 2), (4, 2)} Equations/Inequalities: 1. y = 5x − 3 3. x = |y| + 4 5. y 3 + |x| = 2 √ 2. y 2 + x2 = 5 4. y = x + 2 + 4 6. y < x + 1 Graphs: 1. y 2. y 3. y x x x Math 18 | Page 18  4. y 5. y x x 7.1 Function Defined by an Equation Let X and Y be two sets of real numbers. A function from X to Y can be defined by an equation by giving a method of determining the variable y ∈ Y when the first variable x ∈ X is given. Here, we say that y is a function of x. We often denote functions by letters such as F , G, f , g, K, and so on. If y is a function of x, the name of the function is f , we write y = f (x). The symbol y = f (x) is called the value of f at x, or the image of x under f. The set X is called the domain of the function. The range of f is the set of all possible values of f (x) as x varies throughout the domain, i.e., ranf = {f (x)|x ∈ X}. The symbol that represents an arbitrary number in the domain of a function f is called an independent variable. The symbol that represents an arbitrary number in the range of a function f is called an dependent variable. So, if we write y = f (x), then x is the independent variable and y is the dependent variable. 7.2 Domain Range, and Graph of a Function Definition 7.2.1. Domain the set composed of the first components of all ordered pairs in a function. the set of all independent variable in a function. Definition 7.2.2. Range the set composed of the second components of all ordered pairs in a function. the set of all possible values for the dependent variable in a function. Remarks. In finding the domain of a function, note that P 1. In any fraction , the denominator Q is never allowed to be zero. Q √ 2. In n a, where n is even, it is required that a ≥ 0. Definition 7.2.3. Evaluating a Function To evaluate a function, say f , at a number, we substitute the number for x in the definition of f. Math 18 | Page 19 Example. Let f (x) = 2x2 + x − 1. Evaluate each function value.   1. f (−3) 1 2. f 2 Definition 7.2.4. Graph of a Function If f is a function, then the graph of f is the set of all points (x, y) in the plane R2 for which (x, y) is an ordered pair in f. Example. 4 y f (x) = x3 + 1 2 x −2 2 4 −2 −4 8 Types of Functions 8.1 Polynomial Functions Definition 8.1.1. Polynomial Function A polynomial function of degree n is a function of the form P (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 where n is a nonnegative integer. Remarks. The numbers a0 , a1 , a2 ,... , an are called coefficients of the polynomial. The number a0 is called the constant coefficient or constant term. The number an , the coefficient of the highest power, is the leading coefficient, and the term an xn is the leading term. Example. Which of the following represents a polynomial function? 1. f (x) = 5x4 − x2 + 1 √ 2. h(x) = x2 + x − 3 3. g(x) = 2x(x − 1) 2 4. y = x3 − +5 x x+3 5. G(x) = 2 6. F (x) = 2 Math 18 | Page 20 8.1.1 Constant Function Definition 8.1.2. Constant Function (y = k) Graph is a horizontal line passing through y = k Domain: R Range: {k} x-intercept: none if k ̸= 0, or R if k = 0 y-intercept: k Example. Find the domain, range, intercepts of the function defined by f (x) = 5. Sketch its graph. 8.1.2 Linear Function Definition 8.1.3. Linear Function (y = mx + b, m ̸= 0) Graph is a straight line – To graph, assign at least two values or obtain at least 2 points. Domain: R Range: R x-intercept: mx + b = 0 (solve for x) y-intercept: b Example. Find the domain, range, intercepts of the function y = 2x − 4. Sketch the graph. 8.1.3 Quadratic Function Definition 8.1.4. Quadratic Function (y = ax2 + bx + c, a ̸= 0) Graph is a parabola opening upward (a > 0) or downward (a < 0) y y x x Figure 1: * Figure 2: * a>0 a 0 : y y ≥ or , +∞ 4a  4a 4ac − b2 4ac − b2    a 0 : [d, +∞) Math 18 | Page 22  a < 0 : (−∞, d] √ x-intercept: a bx − c + d = 0 (solve for x) y-intercept: Let x = 0 and solve for y To graph, assign values for points in the domain especially for the endpoint. Example. Find the domain, range, intercepts of each function. Sketch the graph. √ √ 1. f (x) = 3x − 2 + 4 2. h(x) = 5 − 2 1 − 2x √ √ 8.2.2 Square Root Function of the form y = a2 − x2 or y = − a2 − x2 √ √ Definition 8.2.2. Square Root Function of the form y = a2 − x2 or y = − a2 − x2 Graph is a semi-circle with radius a Domain: [−a, a] √ Range: [0, a] for y = a√ 2 − x2 , and [−a, 0] for y = − a2 − x2 x-intercepts: ±a √ y-intercept: a for y = √a2 − x2 , and −a for y = − a2 − x2 √ Example. Find the domain, range, intercepts of the function g(x) = 9 − x2. Sketch the graph. √ √ 8.2.3 Square Root Function of the form y = x2 − a2 or y = − x2 − a2 √ √ Definition 8.2.3. Square Root Function of the form y = x2 − a2 or y = − x2 − a2 Graph is a half-hyperbola Domain: (−∞, −a] ∪ [a, +∞) √ Range: [0, +∞) for y = √x2 − a2 , and (−∞, 0] for y = − x2 − a2 x-intercepts: ±a y-intercept: none √ Example. Find the domain, range, intercepts of the function f (x) = − x2 − 4. Sketch the graph. Math 18 | Page 23 9 Operations on Functions, Inverse of a Function 9.1 Operations on Functions Definition 9.1.1. Algebra of Functions Let f and g be two functions of x, and Df and Dg be the domain of f and g, respectively. Then 1. (f + g)(x) = f (x) + g(x) (Sum) 2. (f − g)(x) = f (x) − g(x) (Difference) 3. (f · g)(x) = f (x) · g(x) (Product)   f f (x) 4. (x) = , g(x) ̸= 0 (Quotient) g g(x) Remarks. The domain of f + g, f − g, and f · g is Df ∩ Dg. f The domain of is Df ∩ (Dg − {x|x ∈ Dg , g(x) = 0}). g Definition 9.1.2. Composition of Functions Let f and g be two functions of x, and Df and Dg be the domain of f and g, respectively. Then the composite function f ◦ g is defined by (f ◦ g)(x) = f (g(x)). Remarks. The domain of f ◦ g is {x|x ∈ Dg , g(x) ∈ Df }. Meaning, the domain of a composition is equal to the intersection of the domain of the inner function and the domain of the resulting function. √ Example. Given that f (x) = x2 − 1 and g(x) = x + 1, find the functions listed below. Determine their domain.   1. (f + g)(x) f 4. (f ◦ g)(x) 3. (x) g 2. (f · g)(x) 5. (g ◦ f )(x) 9.2 Inverse of a Function Definition 9.2.1. One-to-One Function A function f is said to be one-to-one if and only if for any two numbers x1 and x2 , x1 ̸= x2 in the domain of f , then f (x1 ) ̸= f (x2 ). In another way, if f (x1 ) = f (x2 ), then x1 = x2. Example: Non-example: 1. f (x) = 3x + 2 1. h(x) = |x| 2. g(x) = x3 2. m(x) = x2 Definition 9.2.2. Horizontal Line Test A function is one-to-one if and only if every horizontal line intersects the graph of the function in no more than one point. Math 18 | Page 24  Definition 9.2.3. Inverse of a Function If f is a one-to-one function defined by the set of ordered pairs (x, y), then there is a function f −1 , called the inverse of f defined by the set of ordered pairs (y, x). That is, f −1 = {(y, x)|(x, y) ∈ f }. In equation form, x = f −1 (y) if and only if y = f (x). The domain of f −1 is the range of f and the range of f −1 is the domain of f. Remarks. The graph of f and f −1 is symmetric with respect to the line y = x, that is, the graph of f −1 is the reflection of the graph of f with respect to the line y = x. f (f −1 (x)) = x, x is in the domain of f −1 and f −1 (f (x)) = x, x is in the domain of f. Definition 9.2.4. How to Find the Inverse of a Function 1. Let y = f (x). Then we solve for x in terms of y. 2. Obtain x = f −1 (y). 3. Interchange the variables x and y to retain x as the independent variable and y as the dependent variable. Example. Determine if the function is one-to-one. If the function is one-to-one, find the equation of its inverse. Specify the domain and the range of the function and its inverse. Sketch the graph of the function together with its inverse. √ 1. f (x) = 2x 2. f (x) = x − 1 COPYRIGHT NOTICE This material has been reproduced and communicated to you by or on behalf of University of the Philippines pursuant to PART IV: The Law on Copyright of Republic Act (RA) 8293 or the ‘‘Intellectual Property Code of the Philippines’’. The University does not authorize you to reproduce or communicate this material. The Material may contain works that are subject to copyright protection under RA 8293. Any reproduction and/or communication of the material by you may be subject to copyright infringement and the copyright owners have the right to take legal action against such infringement. Math 18 | Page 25 Exercises 1. Prove that for any real number a, a · 0 = 0. (Hint: Start with 0 + 0 = 0) c a 2. Prove that for any real numbers a, b, c with b ̸= 0, if a + b = c, then = + 1. b b 3. Prove that (−1)(−1) = 1. b 4. Prove that = b. 1 b 1 a 5. Prove that if ̸= 0 then b =. a a b b d 6. Prove that if = then b = d. a a −b b 7. Prove that =−. a a 8. Express the following sets in interval notation. a. {x ∈ R|x > 2 or x ≤ 2} e. {x ∈ R|x < 5 or x < 4} b. {x ∈ R|x < 5 and x > −1} f. {x ∈ R|x < 5 and x < 4} c. {x ∈ R|x > −3 or x < −4} d. {x ∈ R|x ≥ 2 and x ≤ 2} g. {x ∈ R|x ≥ 10 and x > 9} 9. Express the following sets in interval notation. a. {x ∈ R|x > 2 or x ≤ 2} e. {x ∈ R|x < 5 or x < 4} b. {x ∈ R|x < 5 and x > −1} f. {x ∈ R|x < 5 and x < 4} c. {x ∈ R|x > −3 or x < −4} d. {x ∈ R|x ≥ 2 and x ≤ 2} g. {x ∈ R|x ≥ 10 and x > 9} 10. Find the solution of the following inequalities. Express your answers in interval notation. a. 0 < 3(x + 7) ≤ 20 b. 4 + 3x > x2 c. x2 − 6x + 18 < 9 d. 4 − 4x > −x2 5 3 e. > x−6 x+2 NOTE: You can’t perform cross multiplication since you don’t know the signs of x − 6 and x + 2. x+3 f. > −3 5 g. |x − 2| ≤ 0 11. Show that the rectangle with vertices A(−3, −2), B(3, −2), C(3, 4), and D(−3, 4) is also a square. 12. Find the perimeter of the quadrilateral with vertices A(−4, −3), B(0, −7), C(6, −1), and D(2, 3). Math 18 | Page 26 13. Show that the triangle is isosceles, given its vertices are A(−1, 3), B(0, 0), and C(1, 3). 14. Show that the points A(−4, −1), B(−2, −3), C(4, 3), and D(2, 5) are vertices of a rectangle. 15. Show that the following points are vertices of a parallelogram: A(6, 1), B(5, 6), C(−4, 3), and D(−3, −2). 16. Prove that the points are vertices of a right triangle: A(0, −7), B(−4, −3), and C(6, −1). 17. Show that the triangle whose vertices are A(2, 1), B(5, 5), and C(−2, 4) is an isosceles triangle. 18. Show that the points A(2, 1), B(5, 3), and C(−1, −5) are collinear. 1 19. Find the general equation of the line that passes through the point (5, 1) and has slope. 4 20. Find the general equation of a line having x-intercept 3 and y-intercept −1. 21. Find the general equation of the line passing through point (−3, 2) and the point of intersection of lines y = 3x − 2 and y = −4x + 5. 22. Two perpendicular lines L1 and L2 share the same x-intercept point (3, 0). If L1 passes through the y-axis when y = 9, what is the y-intercept of L2 ? 23. Suppose two perpendicular lines L1 and L2 intersect at P (4, 4), where mL1 > 0. If the distance of the x-intercept of L1 to P is 5 units, what is the general equation of the line L2 ? 24. Find the domain, range, intercepts of each function. Sketch the graph. √ a. f (x) = −3 3x + 3 + 3 d. k(x) = 5 + 2x 2 √ b. g(x) = x + 2x + 2 e. l(x) = − x2 − 1 √ √ c. h(x) = 16 − x2 f. m(x) = 3 x + 4 − 3 25. Determine if the function is one-to-one. If the function is one-to-one, find the equation of its inverse. Specify the domain and the range of the function and its inverse. Sketch the graph of the function together with its inverse. a. f (x) = −3x + 1 √ b. f (x) = 2 2 − x Math 18 | Page 27