6.4 Use Inverse Functions
Inverse
relation
A relation that interchanges
the input and output values of the original relation
Inverse
function
The original relation and its
inverse relation whenever both relations are functions
Example 1
Find an equation for the
inverse of the relation y = 7x - 4.
y = 7x
- 4 Write original equation.
x = 7 y - 4 Switch x and y.
x + 4 = 7
y Add 4 to
each side.
+ = x y 1 7 4 7
Solve for y.
This is the inverse relation.
INVERSE
FUNCTIONS
Functions f and g are
inverses of each other provided:
f(g(x))= __x__ and g(f(x))
= __x__
The function g is
denoted by f-1, read as
“f inverse.”
Example 2
Verify that f(x)
= 7x - 4
and f-1(x)
=
are inverses.
|
Show that f(f-1(x))
= x. |
Show that f (f-1(x))
= x. |
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f-1(f(x))
= f-1(7x
- 4) |
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x = + æ è ç ö ø ÷ - 7 1 7 4 7 4 |
4 4 x = - + 1 7 ( 7 4 ) 4 7 |
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= x + 4 - 4 |
7 x = - + 7
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=
__x__ |
=
__x__ |
Example 3
Find the inverse of a
power function
Find the inverse of f (x)
= 4x2, x < 0. Then graph f and f-1.
f(x)
= 4x2 Write
original function.
y = 4x2 Replace f(x)
with y.
1
x
= 4y2 Switch
x and y.
1 = x y 2 4
Divide each side by 4.
± = x y 2
Take square roots of each side.
x 1 2 .)
The domain of f is restricted
to negative values of x. So, the range of f-1 must also
be restricted to negative values, and therefore the inverse is f-1(x)
= 
.(If
the domain were restricted to x ³ 0, you would choose f -1(x)
=
HORIZONTAL LINE TEST
The inverse of a function f
is also a function if and only if no horizontal line intersects the graph
of f __more than once__ .
Function Not
a function
Example 4
Find the inverse of a
cubic function
Consider the function .
Determine whether the inverse of f is a function. Then find the inverse.
Solution
Graph the function f. Notice
that no __horizontal line__ intersects the graph more than once. So, the
inverse of f is itself a __function__. To find an equation for f-1,
complete the following steps.
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Write original function. |
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Replace f(x)
with y. |
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x y = + 3 1 4 3
- = 1 |
Switch x and y. |
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x y 3 3 4 |
Subtract __3__ from
each side. |
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- = x y 3 4 12 |
Multiply each side by __4__. |
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Take cube root of each
side. |
The inverse of f is f
-1(x) = 