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Inverse Functions
Goals p Find inverses of linear functions. p Find inverses of nonlinear functions.
VOCABULARY
Inverse relation A relation that maps the output values of an original relation back to their original input values Inverse functions A relation and its inverse relation whenever both relations are functions
Your Notes
Example 1 Finding an Inverse Relation
Find an equation for the inverse of the relation y 3 x 1. Solution y 3 x 1 Write original equation. x 3 y 1 Switch x and y. x 1 3 y Add 1 to each side.
^1 3
x ^1 3
y Divide each side by 3.
The inverse relation is y ^1 3
x ^1 3
INVERSE FUNCTIONS
Functions f and g are inverse functions of each other provided: f(g(x)) x and g(f(x)) x The function g is denoted by f^1 , read as โf inverse.โ
162 Algebra 2 Notetaking Guide โข Chapter 7
Your Notes
Verify that f(x) 3 x 1 and f^1 (x)
x
are inverses.
Solution Show that f(f^1 (x)) x and f^1 (f(x)) x.
f(f^1 (x)) f^1 3
x ^1 3
f^1 (f(x)) f^1 (3x 1)
(^3) ^1 3
x ^1 3
1 ^1 3
(3x 1) ^1 3
x 1 1 x
x x
Example 2 Verifying Inverse Functions
Find the inverse function of the function f(x) ^1 4
x^2 , x โค 0.
Solution f(x)
x^2 Write original function.
y ^1 4
x^2 Replace f(x) with y.
x ^1 4
y^2 Switch x and y.
4 x y^2 Multiply each side by 4. 2 x y Take square roots of each side.
Because the domain of f is restricted to nonpositive values, the inverse function is f^1 (x) 2 x . (You would choose f^1 (x) 2 x if the domain had been restricted to x โฅ 0.) Check To check your work, graph f and f^1. Note that the graph of f^1 (x) 2 x is the reflection of the graph of f(x) ^1 4
x^2 , x โค 0 in the line y x.
1 3
3 1
3
3 (^1) 1 x
y f(x) x 2 x โค 0
f ^1 (x) 2 x
1 4
Example 3 Finding an Inverse Power Function
Lesson 7.4 โข Algebra 2 Notetaking Guide 163
