Pc Inverse Functions Intro

INVERSE FUNCTIONS By MATH PROJECT

Inverse Functions OBJECTIVES The Inverse of a Function Find inverse functions informally and verify that two functions are inverses of each other The Graph of the Inverse of a Function Use graphs of functions to decide whether functions have inverses Finding the Inverse of a Function Algebraically How to find inverse functions algebraically

Consider the graph of the Graphical Approach: quadratic equation y = x2 . Is this a function? It passes the vertical line test. Does the graph have an inverse? It fails the horizontal line test. Consider the graph of y = x 2 only when x ≥ 0 x y

0 0

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2 4

3 9

4 16

y

Vertex (0, 0) x

Consider the graph of y = x 2 only when x y

0 0

1 1

2 4

3 9

4 16

x≥0

y

Is this a function? It passes the vertical line test. Does the graph have an inverse? It passes the horizontal line test.

x

y=x2 Switch x and y . x=y2 Solve for y by applying the square root property.

x=

y

2

x=y What is the graph? x y

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4 2

9 3

16 4

What is the graph? x y

0 0

1 1

4 2

9 3

16 4

Identity function y=x

Finding the Inverse of a Function Algebraically     

Use the Horizontal Line Test to decide whether f has an inverse. In the equation for f (x), replace f (x) by y. Interchange the roles of x and y, solve for y. Replace y by f –1(x) in the new operation. Verify that f and f –1 are inverse of each other by showing that the domain of f must be equal to the range of f –1, the range of f must be equal to the domain of f –1 , and f ( f –1 (x) ) = f –1( f (x) )

The Inverse of a Function Let f and g be the 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 f. The function g is the inverse of the function f , denoted f –1. So, f ( f –1(x) ) = x f –1( f (x) ) = x The domain of f must be equal to the range of f –1 , and the range of f must be equal to the domain of f –1.

Horizontal Line Test for the Inverse Function A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point.