Inverse Functions PDF

Key Concepts of Inverse Functions, Exponential Functions and Logarithmic Functions - Part 001  Objectives At the end of this lesson, the learner should be able to: 1. Represent real-life situations using one-to one functions 2. Determine the inverse of a one-...

Key Concepts of Inverse Functions, Exponential Functions and Logarithmic Functions - Part 001  Objectives At the end of this lesson, the learner should be able to: 1. Represent real-life situations using one-to one functions 2. Determine the inverse of a one-to-one function 3. Represent an inverse function through its (a) table of values, and (b) graph 4. Find the domain and range of an inverse function Represent real-life situations using one-to one functions  Definitions Domain The set of all values of x for which the function is defined Range The set of all values of y for which the function takes from the values of the domain  One-to-One function Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.  One-to-One Function A function f is one-to one if for and and b in its domain, f(a) = f(b) implies a = b. Other – each component of the range is unique.  Horizontal Line Test A test for one-to one If a horizontal line intersects the graph of the function in more than one point, the function is not one-to one  Example Image taken from http://www.math.toronto.edu/preparing-for-calculus/4_functions/we_3_one_to_one.html  Example: Circumference of a Circle The circumference function for a circle is a one-to-one function as no two values for diameter (d) would give the same value for the circumference 𝐶 = 𝜋𝑑  Example: Price you pay for goods you buy If you are buying apples at Php 28.00 per piece, the function that represents the amount you need to pay when you buy x number of apples is 𝑃𝑟𝑖𝑐𝑒 𝑡𝑜 𝑃𝑎𝑦 = 𝑃ℎ𝑝 28.00 (𝑥) where x is the number of apples you will buy This is one-to-one since no two values of x will give you the same amount you need to pay. Determine the inverse of a one-to-one function Represent an inverse function through its (a) table of values, and (b) graph http://college.cengage.com/mathematics/shared/content/digital_lessons/inverse_functions.ppt  Existence of an Inverse Function A function f has an inverse function if and only if f is one to one.  Find an Inverse Function 1. Determine if f has an inverse function using horizontal line test. 2. Replace f(x) with y 3. Interchange x and y 4. Solve for y 5. Replace y with 1 f ( x)  Definition of Inverse Function Let f and g be two functions such that f(g(x))=x for every x in the domain of g and g(f(x))=x for every x in the domain of. g is the inverse function of the function f  Definition A function y = f (x) with domain D is one-to-one on D if and only if for every x1 and x2 in D, f (x1) = f (x2) implies that x1 = x2. A function is a mapping from its domain to its range so that each element, x, of the domain is mapped to one, and only one, element, f (x), of the range. A function is one-to-one if each element f (x) of the range is mapped from one, and only one, element, x, of the domain.  Horizontal Line Test y (0, 7) (4, 7) Horizontal Line Test y=7 A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) 2 in more than one point. x 2 Example: The function y = x2 – 4x + 7 is not one-to-one on the real numbers because the line y = 7 intersects the graph at both (0, 7) and (4, 7).  Example Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. a) y = x3 b) y = x3 + 3x2 – x – 1 y y 8 8 4 4 -4 4 -4 4 x x one-to-one not one-to-one  Inverse Functions Function y = |x| + 1 Inverse relation x = |y| + 1 x y x y 2 2 1 3 1 3 0 2 0 2 -1 1 -1 1 -2 -2 Domain Range Range Domain Every function y = f (x) has an inverse relation x = f (y). The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}. x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}. The inverse relation is not a function. It pairs 2 to both -1 and +1.  Ordered Pairs The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation. Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}. The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}. The domain of the inverse relation is the range of the original function. The range of the inverse relation is the domain of the original function.  The graphs of a relation and its inverse are reflections in the line y = x. Example: Find the graph of the inverse relation geometrically from the graph of f (x) = y y=x ( x  2) 3 2 4 The ordered pairs of f are given by the equation y  ( x 3  2). x 4 -2 2 The ordered pairs of the inverse are given by ( y  2) 3 -2 ( x 3  2) ( y 3  2) x y x 4 4 4  Inverse Relation: Algebraically Example: Find the inverse relation algebraically for the function f (x) = 3x + 2. y = 3x + 2 Original equation defining f x = 3y + 2 Switch x and y. 3y + 2 = x Reverse sides of the equation. ( x  2) y= Solve for y. 3 To calculate a value for the inverse of f, subtract 2, then divide by 3.  Inverse Function For a function y = f (x), the inverse relation of f is a function if and only if f is one-to-one. For a function y = f (x), the inverse relation off is a function if and only if the graph of f passes the horizontal line test. If f is one-to-one, the inverse relation of f is a function called the inverse function of f. The inverse function of y = f (x) is written y = f -1(x).  Example Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph. y y = f -1(x) The graph of f passes the horizontal line test. y=x The inverse y = f(x) relation is a function. x Reflect the graph of f in the line y = x to produce the graph of f -1.  Composition of a Function The inverse function is an “inverse” with respect to the operation of composition of functions. The inverse function “undoes” the function, that is, f -1( f (x)) = x. The function is the inverse of its inverse function, that is, f ( f -1(x)) = x. Example: The inverse of f (x) = x3 is f -1(x) = 3 x. 3 3 f -1( f(x)) = x = x and f ( f -1(x)) = (3 x )3 = x.  Composition of Functions x 1 Example: Verify that the function g(x) = is the inverse of f(x) = 2x – 1. 2 ( f ( x)  1) ((2 x  1)  1) 2x g( f(x)) = = = =x 2 2 2 f(g(x)) = 2g(x) – 1 = 2( x  1 ) – 1 = (x + 1) – 1 = x 2 It follows that g = f -1. Find the domain and range of an inverse function  Inverse Function g is the inverse of f if the domains and ranges are interchanged. f = {(1,2),(3,4), (5,6)} g= {(2,1), (4,3),(6,5)} 1 g ( x)  f ( x)  Inverse of a function f  1, 2  ,  3, 4  ,  5,6  f 1   2,1 4,3 ,  6,5   Inverse of function f  1, 2  ,  3, 2  ,  5, 2  f 1   2,1 ,  2,3 ,  2,5 