1.
4-5 INVERSE FUNCTIONS
Objectives:
1. Find the inverse of a function, if it exists.
2.
INVERSE EXAMPLE
Conversion
formulas come in pairs, for
example:
These
formulas “undo” each other, so
they are inverses.
3.
DEFINITION OF INVERSES
Two
functions f and g are inverses if:
f(g(x)) = x and g(f(x)) = x
To
check if two functions are inverses,
perform both compositions and make
sure both equal x.
4.
EXAMPLE 1
If
and
show that f and g are inverses.
6.
INVERSE NOTATION
The
inverse of f is written f -1
f -1(x)
Note:
is the value of f -1 at x
is not
7.
FINDING INVERSES
graph of f -1 is the reflection of f
over the line y = x
Can be found by
switching x and y
in the ordered pairs.
Find equation of f -1
by switching x and y
in the equation and
solving for y.
The
8.
EXAMPLE 2
f(x) = 4 – x2 for x 0.
Sketch the graph of f and f -1 (x)
Find a rule for f -1 (x)
Let
9.
YOU TRY!
g(x) = (x – 4)2 – 1 for x 4.
Sketch the graph of g and g -1 (x)
Find a rule for g -1 (x)
Let
10.
EXAMPLE 3
Suppose
a function f has an inverse.
If f(2) = 3, find:
f -1
(3)
f(f -1(3))
f -1(f(2))
11.
YOU TRY!
Suppose
a function g has an inverse.
If g(5) = 1, find:
g -1
(1)
g -1(g(5))
g(g -1(1))
12.
DO ALL FUNCTIONS HAVE
INVERSES?
the graph of y = x2 over the
line y = x.
Is the result a function?
Reflect
13.
ONE-TO-ONE
Only
functions that are one-to-one
have inverses.
One-to-one means each x value has
exactly one y value and each y has
exactly one x
Can check using horizontal line test.
14.
EXAMPLE 4
Which
functions are one-to-one?
Which have inverses?
15.
YOU TRY!
Is h(x) one-to-one?
Does it have an inverse?
16.
EXAMPLE 5
State
whether each function has an
inverse. If yes, find f -1 (x) and show
f(f -1(x)) = f -1(f(x)) = x
17.
YOU TRY!
Does
have an
inverse? If so, find f -1 (x).
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