Functions PDF

FUNCTIONS FUNCTIONS AND RELATIONS What is a Relation? A Relation is any set of ordered pairs. Example : (1, 2) (-2, 0) (3, -5) (-4, -4) (0, 7) What is a Function? A Function is a Relation in which each unique member of the domain is matched with exactly one member of the range. Dom...

FUNCTIONS FUNCTIONS AND RELATIONS What is a Relation? A Relation is any set of ordered pairs. Example : (1, 2) (-2, 0) (3, -5) (-4, -4) (0, 7) What is a Function? A Function is a Relation in which each unique member of the domain is matched with exactly one member of the range. Domain – set of all first members (coordinates or abscissa) in a Relation. Range – set of all second members (ordinates or mantissa) in a Relation. In the above example, D = {1, -2, 3, -4, 0} R = {2, 0, -5, -4, 7} FUNCTIONS VS RELATIONS How do you tell whether the Relation is a Function or not? A. By definition : 1. One-to-one Relation – one unique x-value for one unique y-value. Example : (1, 2) (-3, 0) (-4, 5) (0, -9) (-1, -1) Function 2. One-to-many Relation – repeating x-value for one unique y-value. Example : (1, 2) (-3, 0) (1, 5) (0, -9) (-3, -1) Not a Function 3. Many-to-one Relation – one unique x-value for repeating y-value. Example : (1, 2) (-3, 2) (-4, 5) (0, -9) (-1, 2) Function 4. Many-to-many Relation – repeating x-value for repeating y-value. Example : (1, 2) (1, 0) (-4, 2) (0, 2) (1, -1) Not a Function B. By Vertical Line Test (used in graphs) If a vertical line is drawn on the graph, and the line intersects the graph at one and only one point, then the graph is a Function. If a vertical line is drawn on the graph, and the line intersects the graph at more than one point, then the graph is a Not a Function. Examples : Function Function Not a Function  Not a Function Function Not a Function TYPES OF FUNCTIONS : A. Linear Functions B. Absolute Value Functions C. Quadratic Functions D. Cubic Functions E. Inverse Functions F. Rational Functions G. Piecewise Functions H. Exponential Functions I. Logarithmic Functions A. Linear Functions - are those whose graph is a straight line where x is the independent variable and y is the dependent variable. Examples : y = 2x – 1 3y = x + 2 -5x + 6y – 4 = 0 B. Absolute Value Functions - are those that contain algebraic expressions within absolute value symbols. Examples : y = |2x – 3| 2y = |6 + 4x| y = |-3x| C. Quadratic Functions - polynomial functions with one or more variables in which the highest degree term is of the second degree. Examples : y = x2 3y = x2 – 4 y = -5(x + 1) + 2 D. Cubic Functions - polynomial functions with one or more variables in which the highest degree term is of the third degree. Examples : y = 2x3 – 4x2 + x – 2 2y = x3 + 6x – 1 y = -7x3 + x2 – 2  E. Inverse Functions - functions that reverses other functions. Examples : y = x + 2 ; y-1 = x – 2 y = 3x + 2 ; y-1 = (x – 2)/3 4y = x – 1 ; y-1 = 4x + 1 F. Rational Functions - functions that can be written as ratios of two polynomial functions. Examples : y = 2/x y = (x + 2)/(x – 3) y = (4x + 1)/(6 + 5x) G. Piecewise Functions - functions defined by multiple sub-functions, each sub-function applying to a certain interval of the main function’s domain, sub-domain. Example : x2 – 5x , x ≤ -10 y= x + 12 , -10 < x < -2 x3/(x – 4) , x ≥ -2 H. Exponential Functions - functions in which the independent variable occurs as an exponent. Examples : y = ex y = 3ex + 4 y = -2e5x - 1 I. Logarithmic Functions - functions that are inverses of exponential functions. Examples : y = log x y = log2(x – 2) y = ln(3x + 4) EVALUATION OF FUNCTIONS 1. Given : f(x) = x + 2 find f(x) if a. x = 3 b. x = -5 c. x = -1 d. x = ¼ e. x = 23 Solution : a. f(3) = 3 + 2 f(3) = 5 b. f(-5) = -5 + 2 f(-3) = -3 c. f(-1) = -1 + 2 f(-1) = 1 d. f(¼ ) = ¼ + 2 f(¼) = 9/4 e. f(23) = 8 + 2 f(23) = 10  2. Given : f(x) = 2x3 – 3x + 1 find f(x) if a. x = 1 b. x = -1 c. x = 2 d. x = -2 e. x = ½ 3. Given : f(x) = 3x – 4 find f(x) if x+1 a. x = -2 b. x = 0 c. x = 3 d. x = -½ e. x = 1 4. Given : f(x) = 23x – 1 find f(x) if a. x = 0 b. x = -1 c. x = 1 d. x = 3 e. x = -2 OPERATIONS ON FUNCTIONS ADDITION/SUBTRACTION OF FUNCTIONS Given : f(x) = x2 – 2x + 2 g(x) = 3x2 + 4x – 1 h(x) = 6x – x2 – 5 Find the following : a. (f + g)(x) = (x2 – 2x + 2) + (3x2 + 4x – 1) f. (g – f)(-1) = x2 – 2x + 2 + 3x2 + 4x – 1 g. (f – h)(0) = 4x2 + 2x + 1 h. (h + g)(-2) b. (f – g)(x) = (x2 – 2x + 2) – (3x2 + 4x – 1) i. (f + g – h)(1) = x2 – 2x + 2 – 3x2 – 4x + 1 j. (g – h – f)(2) = -2x2 – 6x + 3 c. (h – g)(x) d. (f + h)(x) e. (g – h)(x) MULTIPLICATION/DIVISION OF FUNCTIONS Given : f(x) = x + 2 g(x) = 3x – 4 h(x) = 4x2 – 2x – 20 Find the following : a. (f ⋅ g)(x) = (x + 2)(3x – 4) = 3x2 – 4x + 6x – 8 = 3x2 + 2x – 8  b. (f ⋅ h)(x) = (x + 2)(4x2 – 2x – 20) = 4x3 – 2x2 – 20x + 8x2 – 4x – 40 = 4x3 + 6x2 – 24x – 40 c. (h ÷ f)(x) = (4x2 – 2x – 20)/(x + 2) = 4x – 10 d. (g ⋅ f)(-1) e. (g ⋅ h)(0) Given : f(x) = x – 3 g(x) = 1 + 2x h(x) = 6x2 – x – 2 a. (f ⋅ g)(x) = b. (f ⋅ h)(x) = c. (h ÷ g)(x) = d. (g ⋅ f)(0) = e. (g ⋅ h)(-1) = COMPOSITION OF FUNCTIONS Given : f(x) = x + 2 g(x) = 3x – 4 h(x) = 4x2 – 2x – 20 Find the following : a. (f g)(x) = (3x – 4) + 2 d. (g h)(x) = 3x – 4 + 2 e. (g f)(x) = 3x – 2 f. (f g)(0) b. (f h)(x) = (4x2 – 2x – 20) + 2 g. (f h)(1) = 4x2 – 2x – 20 + 2 h. (h f)(-1) = 4x2 – 2x – 18 i. (g h)(2) c. (h f)(x) = 4(x + 2)2 – 2(x + 2) – 20 j. (h f)(-2) = 4(x2 + 4x + 4) – 2x – 4 – 20 = 4x2 + 16x + 16 – 2x – 4 – 20 = 4x2 + 14x – 8 Given : f(x) = x + 1 g(x) = 2x – 3 h(x) = x2 + 4x – 5 Find the following : a. (f g)(x) = f. (f g)(0) = b. (f h)(x) = g. (f h)(1) = c. (g f)(x) = h. (g f)(-1) = d. (h g)(x) = i. (h g)(2) = e. (h f)(x) = j. (h f)(-2) =