General Mathematics Past Paper PDF

GENERAL MATHEMATICS LESSON 1: RELATIONS AND FUNCTIONS RELATIONS REPRESENTING FUNCTIONS A rule that relates values from a set of Set of ordered pairs - there should be no values (called t...

GENERAL MATHEMATICS LESSON 1: RELATIONS AND FUNCTIONS RELATIONS REPRESENTING FUNCTIONS A rule that relates values from a set of Set of ordered pairs - there should be no values (called the domain) to a second set repeating x-value of values (called the range). Table of values - there should be no repeating x-value FUNCTION Mapping diagram - one-to-one, many-to- A set of ordered pairs (x,y) such that no two one function ordered pairs have the same x-values but Graph - Vertical line test (drawing a vertical different y-values. line must only intersect the graph once) Using functional notation, we can write Equations - no negative, decimal, or f(x) = y read as "f of x is equal to y." fractional exponent ACTIVITY: RELATIONS VS. FUNCTIONS Determine whether the following relation is a function. 1. {(2,4), (6,8), (5,7), (3,5)} 6. x y 7. x y 8. 2. {(1,2), (1,3), (4,5), (6,7)} -1 1 1 4 3. f(x) = x² - 4/x + 5 -2 2 2 3 4. f(x) = √(x) -4 5 3 2 5. y = x³ + 6x² - 3x + 27 -4 6 4 1 GENERAL MATHEMATICS LESSON 2: EVALUATING FUNCTIONS AND PIECEWISE FUNCTION EVALUATING FUNCTION REPRESENTING FUNCTIONS To evaluate a given function, substitute the x A piecewise function is a function that has a value to the function and then simplify. different expression for different intervals of the input. ACTIVITY: EVALUATING FUNCTIONS Evaluate the following functions given the set of x values. 1. f(x) = x² - 4x - 4; find f(2) 4. 150x , if 0 < x ≤ 99 2. f(x) = 5x - 25; find f(5) f(x) = 500 , if x = 100 3. f(x) = x³ - 5x² - x - 11; find f(3) 500 + 100 (x - 100) , if x > 100 a. Find f(100) b. Find f(9) c. Find f(102) GENERAL MATHEMATICS LESSON 3: FUNCTIONS OPERATIONS FUNCTIONS OPERATIONS (f + g)(x) = f(x) + g(x) Let f and g be functions defined at a real number x. The sum, difference, product, and (f - g)(x) = f(x) - g(x) quotient, of f and g are the functions defined by: (f g)(x) = f(x) g(x) (f ÷ g)(x) = f(x) ÷ g(x), g(x) ≠ 0 ACTIVITY: OPERATIONS ON FUNCTIONS Solve the following operations on functions. 1. Functions of f, g, and h are given by a. (g + h)(x) e. (g + h)(2) f(x) = x² + x - 12 b. (f - g)(x) f. (f - g)(3) g(x) = 5x - 2 c. (g h)(x) g. (g h)(4) h(x) = x - 3 d. (f ÷ h)(x) h. (f ÷ h)(5) GENERAL MATHEMATICS LESSON 4: FUNCTION COMPOSITION FUNCTION COMPOSITION (f ∘ g)(x) = f[g(x)] Let fng be functions, the composite of functions f and g is defined as (f ∘ g)(x) = f[g(x)]. The domain of (f ∘ g)(x) consists of all real numbers f in the domain of g for which g(x) is in the domain of f. ACTIVITY: FUNCTION COMPOSITION Find the composite of the given function. 1. Functions of f, g, and h are given by a. (f ∘ h)(x) e. (f ∘ h)(5) f(x) = x² + x + 6 b. (f ∘ g)(x) f. (f ∘ g)(4) g(x) = x - 2 c. (g ∘ h)(x) g. (g ∘ h)(3) h(x) = x - 3 d. (h ∘ g)(x) h. (h ∘ g)(2) GENERAL MATHEMATICS LESSON 5: INVERSE OF FUNCTIONS ONE-TO-ONE FUNCTION REPRESENTING FUNCTIONS The function of f is one-to-one f or any x 1 , x 2 , Let f be a one-to-one with domain A and in the domain of f, then f(x 1 ) ≠ f(x 2 ). That is, range B. Then the inverse of f, denoted by f -1 the same y-value is never paired with two is a function with domain Band range A different x-values. defined by f (y) = x if and only if f(x) = y for -1 any y in B A linear function is a one-to-one function. A quadratic function is not a one-to-one A function has an inverse if and only if it is function. one-to-one. HORIZONTAL LINE TEST f(f -1 (x)) = f -1 (f(x)) If every horizontal line intersects the graph of a function f and at most one point, then f is one-to-one. ACTIVITY: INVERSE OF FUNCTIONS Find the inverse of the following functions. 1. {(1,2), (3,4), (5,6), (7,8), (9, 10)} 3. {(1,5), (3,4), (7,6), (8,5), (9, 10)} 2. f(x) = 5x + 3 4. f(x) = ⅔x - 8 GENERAL MATHEMATICS LESSON 6: RATIONAL EQUATIONS AND INEQUALITIES RATIONAL EQUATION Determine if the intervals of the rational expression takes positive or negative values An equation that contains rational a. Determine the x values for which the expressions is referred to as a rational ational expression is zero or undefined. equation. b. Mark the numbers on the number line. c. Select the test points within the interior of RATIONAL EQUATION the interval. e. Summarize the intervals containing the An inequality involving rational expressions. solutions. SOLVING RATIONAL INEQUALITIES Rewrite the inequality as a single rational expression on one side of the inequality symbol and zero on the other side. ACTIVITY: RATIONAL EQUATIONS AND INEQUALITIES Solve the following rational equations and inequalities. 3x 2 7 1 1 4x+2 2x — +— = —- 2. —— +—— = ———- 3. —— ≥1 1. 5 3 30 x-1 x+1 x²-1 x+1 GENERAL MATHEMATICS LESSON 7: RATIONAL FUNCTIONS RATIONAL FUNCTION OBLIQUE ASYMPTOTE A function in the form f(x) = p(x) / q(x), A line of the form ( where p(x) and q(x) are polynomials and y = mx + b which the ( q(x) is not equal to zero. curve approaches but never touches. It exists VERTICAL ASYMPTOTE if the degree of n is larger than the degree A vertical line of the of m, that is, n = m + 1. form x = a which the ( ( curve approaches but never touches. The DOMAIN vertical line passing through the zeros of The set of all values of x that have the denominator of corresponding values of y; it contains all the rational functions values that go into the function. are the vertical asymptotes. RANGE HORIZONTAL ASYMPTOTE The set of all values of y that can be obtained from the possible values of x; it A horizontal line of the contains all possible values of the function. ( form y = b which the ( curve approaches but REPRESENTING FUNCTIONS never touches. To determine the The intersection of the horizontal asymptote graph to the x and y ( ( of a function axis. (0,y) f(x) = p(x)/ q(x), To get the x-intercept, the degrees of the numerator equate the numerator (x,0) and the denominator will be considered. to zero. Let n and m be the degree of p(x) and To get the y-intercept, q(x) respectively. evaluate f(0). a. If nm, there is no horizontal asymptote. same with the x-intercept/s. ACTIVITY: RATIONAL FUNCTION Given the following function, find horizontal and vertical asymptotes, domain, range, x-intercept, y-intercept and zeroes. 2x +3 f(x) = x+2 GENERAL MATHEMATICS ANSWER KEY ACTIVITY: RELATIONS VS. FUNCTIONS Determine whether the following relation is a function. 1. {(2,4), (6,8), (5,7), (3,5)} 6. RELATION 7. FUNCTION 8. RELATION FUNCTION x y x y 2. {(1,2), (1,3), (4,5), (6,7)} -1 1 1 4 RELATION -2 2 2 3 3. f(x) = x² - 4/x + 5 -4 5 3 2 RELATION -4 6 4 1 4. f(x) = √(x) RELATION 5. y = x³ + 6x² - 3x + 27 FUNCTION ACTIVITY: EVALUATING FUNCTIONS Evaluate the following functions given the set of x values. 1. f(x) = x² - 4x - 4; find f(2) 4. 150x , if 0 < x ≤ 99 f(2) = (2)² - 4(2) -4 f(x) = 500 , if x = 100 f(2) = 4 - 8 - 4 500 + 100 (x - 100) , if x > 100 f(2) = -8 a. Find f(100) f(100) = 500 2. f(x) = 5x - 25; find f(5) f(5) = 5(5) -25 b. Find f(9) f(5) = 25 - 25 f(9) = 150(9) f(5) = 0 f(9) = 1350 3. f(x) = x³ - 5x² - x - 11; find f(3) c. Find f(102) f(3) = (3)³ - 5(3)² - (3) - 11 f(102) = 500 + 100(102 - 100) f(3) = 27 - 5(9) - 3 - 11 f(102) = 500 + 100(2) f(3) = 27 - 45 - 3 - 11 f(102) = 500 + 200 f(3) = -32 f(102) = 700 GENERAL MATHEMATICS ANSWER KEY ACTIVITY: OPERATIONS ON FUNCTIONS Solve the following operations on functions. 1. Functions of f, g, and h are given by e. (g + h)(2) f(x) = x² + x - 12 (g + h)(2) = [5(2) - 2] + [(2) - 3] g(x) = 5x - 2 (g + h)(2) = 10 - 2 + 2 - 3 h(x) = x - 3 (g + h)(2) = 7 a. (g + h)(x) f. (f - g)(3) (g + h)(x) = (5x - 2) + (x - 3) (g + h)(x) = 5x - 2 + x - 3 (f - g)(3) = [(3)² + (3) - 12] - [5(3) - 2] (g + h)(x) = 6x -5 (f - g)(3) = 9 + 3 - 12 - 15 + 2 (f - g)(3) = -12 b. (f - g)(x) (f - g)(x) = (x² + x - 12) - (5x - 2) g. (g h)(4) (f - g)(x) = x² + x - 12 - 5x + 2 (g h)(4) = [5(4)- 2][(4) - 3] (f - g)(x) = x² - 4x - 10 (g h)(4) = (20 - 2)(1) (g h)(4) = (18)(1) c. (g h)(x) (g h)(4) = 18 (g h)(x) = (5x - 2)(x - 3) (g h)(x) = 5x² - 15x - 2x + 6 h. (f ÷ h)(5) [(5)² + (5) - 12] (g h)(x) = 5x² - 17x + 6 (f ÷ h)(x) = [5(5) - 2] 25 + 5 - 12 d. (f ÷ h)(x) (f ÷ h)(x) = 25 - 2 (x² + x - 12) (f ÷ h)(x) = 18 (5x - 2) (f ÷ h)(x) = 23 (x + 4)(x - 3) (f ÷ h)(x) = (x - 3) (f ÷ h)(x) = x + 4 GENERAL MATHEMATICS ANSWER KEY ACTIVITY: FUNCTION COMPOSITION Find the composite of the given function. 1. Functions of f, g, and h are given by e. (f ∘ h)(5) f(x) = x² + x + 6 h(5) = (5) - 3 (f ∘ h)(5) = x² + x + 6 g(x) = x - 2 h(5) = 2 (f ∘ h)(5) = (2)² + (2) + 6 h(x) = x - 3 (f ∘ h)(5) = 4 + 2 + 6 a. (f ∘ h)(x) (f ∘ h)(5) = 12 (f ∘ h)(x) = x² + x + 6 (f ∘ h)(x) = (x - 3)² + (x - 3) + 6 f. (f ∘ g)(4) (f ∘ h)(x) = x² - 6x + 9 + x - 3 + 6 g(4) = (4) - 2 (f ∘ g)(4) = x² + x + 6 (f ∘ h)(x) = x² - 5x + 12 g(4) = 2 (f ∘ g)(4) = (2)² + (2) + 6 (f ∘ g)(4) = 4 + 2 + 6 b. (f ∘ g)(x) (f ∘ g)(4) = 12 (f ∘ g)(x) = x² + x + 6 (f ∘ g)(x) = (x - 2)² + (x - 2) + 6 g. (g ∘ h)(3) (f ∘ g)(x) = x² - 4x + 4 + x - 2 + 6 h(3) = (3) - 3 (g ∘ h)(3) = x - 2 (f ∘ g)(x) = x² - 3x + 8 h(3) = 0 (g ∘ h)(3) = (0) - 2 (g ∘ h)(3) = -2 c. (g ∘ h)(x) (g ∘ h)(x) = x - 2 h. (h ∘ g)(2) (g ∘ h)(x) = (x - 3) - 2 g(2) = (2) - 2 (h ∘ g)(2) = x - 3 (g ∘ h)(x) = x - 5 g(2) = 0 (h ∘ g)(2) = (0) - 3 (h ∘ g)(2) = -3 d. (h ∘ g)(x) (g ∘ h)(x) = x - 3 (g ∘ h)(x) = (x - 2) - 3 (g ∘ h)(x) = x - 5 GENERAL MATHEMATICS ANSWER KEY ACTIVITY: INVERSE OF FUNCTIONS Find the inverse of the following functions. 1. {(1,2), (3,4), (5,6), (7,8), (9, 10)} 3. {(1,5), (3,4), (7,6), (8,5), (9, 10)} {(2,1), (4,3), (6,5), (8,7), (10, 9)} No inverse; not a one-to-one function 2. f(x) = 5x + 3 4. f(x) = ⅔x - 8 y = 5x + 3 y = ⅔x - 3 x = 5y + 3 x = ⅔y - 3 5y = x - 3 (3)x = 3(⅔y - 3) 5 5 3x = 2y - 1 x-3 2y = 3x + 1 y= 5 2 2 x-3 3x + 1 f ¹ (x) = y= 5 2 3x + 1 f ¹ (x) = 2 ACTIVITY: RATIONAL EQUATIONS AND INEQUALITIES Solve the following rational equations and inequalities. 3x 2 7 1 1 4x+2 — +— = —- 2. —— +—— = ———- 1. 5 3 30 x-1 x+1 x²-1 3x 2 7 1 1 4x + 2 30 ( —— + —) = —— ( 30) (x - 1) (x + 1 )( ——– + ——–) = ———— (x - 1) (x + 1 ) 5 3 30 x-1 x-1 x² - 1 6(3x) + 10(2) = 7 (x + 1) + (x + 1) = 4x + 2 18x + 20 = 7 2x + 2 = 4x + 2 18x = 7 - 20 4x - 2x = 2 - 2 18x = -13 2x = 0 18 18 x=0 x = -13 / 18 GENERAL MATHEMATICS ANSWER KEY ACTIVITY: RATIONAL EQUATIONS AND INEQUALITIES Solve the following rational equations and inequalities. 2x 3. —— x+1 ≥1 2x x+1 | | | | | | | ——– - ——— ≥ 0 | -2 -1 0 1 2 3 x+1 x+1 x-1 ——– ≥ 0 x < -1 -1 < x < 1 x>1 x+1 x+1 - + + x-1=0 x+1=0 x-1 - - + x=1 x ≠ -1 = + - + {x ∈ ℝ | x < -1 or x ≥ 1} ACTIVITY: RATIONAL FUNCTION Given the following function, find horizontal and vertical asymptotes, domain, range, x-intercept, y-intercept and zeroes. 2x +3 horizontal asymptote: y = 2 x-intercept: (-3/2, 0) f(x) = x+2 vertical asymptotes: x = -2 y-intercept: (0, 3/2) domain: {x | x ≠ -2} zeroes: range: {y | y ≠ 2}