Transcript 1 9 Inverse Functions

Objectives:
1. To find the inverse of
a function
algebraically and
graphically
2. To determine if the
inverse of a function
is a function
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Assignment:
P. 99: 1-7 odd, 13, 19
P. 100: 25, 26, 29-32,
33-53 odd, 55, 59, 63
P. 101: 75, 77
P. 102: 89
Read: P. 103-108
Inverse
Inverse Relation
Inverse Function
Horizontal Line Test
One-to-One Function
You will be able to
find the inverse of a
function algebraically
Let f(x) = 4x + 2 and g(x) = 1/4x – 1/2. Find the
following compositions.
1. f(g(x))
2. g(f(x))
An inverse relation is a relation that switches the
inputs and output of another relation.
Inverse
relations
“undo”
each other
If a relation and its inverse are both functions,
then they are called inverse functions.
f  g ( x)   x and g  f ( x)   x
f -1 = “f inverse” or “inverse of f ”
The inverse of a
function is not
necessarily a function.
If a relation and its inverse are both functions,
then they are called inverse functions.

1

f f ( x)  x and f
1
 f ( x)   x
For f and f -1 to be inverse
functions, 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.
Verify that f(x) = 2x + 3 and f -1(x) = ½x – 3/2 are
inverse functions.
Since the inverse of a function “undoes” the
function, to find an inverse informally:
1. Think of what steps would “undo” your
function.
2. Write new function.
3. Check to see if

1

f f ( x)  x and f
1
 f ( x)   x
Find the inverse of each function informally.
1. f (x) = x + 3
2. f (x) = 2x
3. f (x) = x3
Since the inverse of a function switches the x- and
y-values of the original function, we can easily
find the inverse of a function algebraically:
1. Let f(x) = y, if necessary.
2. Exchange the x and y variables.
3. Solve for y.
1
1
f
f
(
x
)

x
and
f
 f ( x)   x .
4. Check that 

5. Check the domain and range situation.
Find the inverse of 𝑓 𝑥 = −23𝑥 + 2.
Let y = mx + b. Find the inverse of y. What is the
relationship between the slopes of inverse
linear functions?
Find the inverse of the given function.
1. f ( x )  x  4
2. f ( x )  2 x  1
3. f ( x)  3 x  1
For some nonlinear
functions, you have to
first restrict the domain
to find its inverse.
y  x2
ℝ
2
x y
 xy
y  x or y   x
?
For some nonlinear
functions, you have to
first restrict the domain
to find its inverse.
y  x2
0, ∞
2
x y
 xy
y  x or y   x
For some nonlinear
functions, you have to
first restrict the domain
to find its inverse.
y  x2
−∞, 0
2
x y
 xy
y  x or y   x
Find the inverse of the given function.
1 3
1. f ( x)  x , x  0 2. f ( x) 
x
27
6
3. f ( x)   x3  4
You will be able to
find the inverse of a
function (or relation)
graphically
Suppose we drew a
triangle on the
coordinate plane.
Geometrically speaking,
what would happen if
we switched the x- and
the y-coordinates?
 x, y    y, x 
Geometrically
speaking, when you
switch the x- and the
y-coordinates of a
graph, the graph will
reflect across the line
y = x.
This is what happens
with inverses.
Therefore, the graphs of
inverse functions are
reflections across the
line y = x.
Furthermore, to graph
f -1, first graph f and
then switch the xand y- coordinates of
some key points.
Graph each of the following functions, then
graph the inverse by switching the x- and yvalues. Is the inverse a function?
1. y = x2
2. y = √x
3. y = x3
You will be able to
tell if the inverse of a
function is also a
function
As the previous Exercise demonstrated, even
though you can find the inverse of a function,
the inverse itself may not be a function.
Remember, we overcome this shortcoming by
restricting the domain of the original function.
Recall that we can use the vertical line test to
see if a graph represents a function.
The question is:
How can we
tell if a
function’s
inverse will
be a
function?
Function
Not a Function
The inverse of a function f is also a function iff
no horizontal line intersects the graph of f
more than once.
Graph the function f. Then use the graph to
determine whether f -1 is a function.
1 5
1. f ( x)  x
5
2. f ( x)  x  4
3. f ( x)  2 x6 , x  0
If f passes both the
vertical and the
horizontal line tests—
that is, if f and f -1 are
functions—then f is a
one-to-one function.
Every input has
exactly one output
Every output has
exactly one input
Explain why the following functions are not
inverse functions.
f ( x)  x 2
g ( x)  x
Objectives:
1. To find the inverse of
a function
graphically and
algebraically
2. To determine if a
function has an
inverse
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•
•
•
•
Assignment:
P. 99: 1-7 odd, 13, 19
P. 100: 25, 26, 29-32,
33-53 odd, 55, 59, 63
P. 101: 75, 77
P. 102: 89
Read: P. 103-108