Consider the mapping diagram of the function shown below.
Let's now reverse the arrows, as follows:
Now, we have a mapping diagram that "reverses" the action of the function The diagram itself represents a new function, which we call the inverse of the function and denote by
We can evaluate the inverse function at any point in the range of For example, to find the value of we notice that we have an arrow from (on the right) to (on the left). Therefore,
In the original function, that same arrow went from (on the left) to (on the right). So, we have that and as a result, we obtain So, applying the function and then "reversing" the action using will get us back to
In general, this is true for every in the domain of the function
The function is represented by the mapping diagram above. What is
Method 1
The function reverses the action of as depicted below.
From the above, we see that
Method 2
From the given mapping diagram, we have
Applying to both sides of the above equation, and using the fact that we get
Therefore, we conclude that
The function $f(x)$ is represented by the mapping diagram above. What is $f^{-1}(4)?$
|
a
|
$-\dfrac 1 2$ |
|
b
|
$\dfrac 1 4$ |
|
c
|
$2$ |
|
d
|
$-4$ |
|
e
|
$5$ |
The function $f(x)$ is represented by the mapping diagram above. What is $f^{-1}(3)?$
|
a
|
$-\dfrac 1 3$ |
|
b
|
$-3$ |
|
c
|
$2$ |
|
d
|
$-1$ |
|
e
|
$\dfrac 1 3$ |
Given the function defined by the table above, what is the value of
From the table, we have
Applying to both sides of the above equation, and using the fact that we get
Therefore, we conclude that
| $x$ | $-9$ | $-6$ | $-3$ | $0$ | $3$ | $6$ | $9$ |
| $f(x)$ | $9$ | $3$ | $2$ | $0$ | $-2$ | $-9$ | $-20$ |
\[ \]
Given the function $f(x)$ defined by the table above, what is the value of $f^{-1}(-9)?$
|
a
|
$6$ |
|
b
|
$9$ |
|
c
|
$-2$ |
|
d
|
$20$ |
|
e
|
$-9$ |
Given that \[ f(0) = 3, \quad f(3) = 6, \quad f(6) = 0, \]
find the value of $f^{-1}(3).$
|
a
|
$6$ |
|
b
|
$-6$ |
|
c
|
$-3$ |
|
d
|
$0$ |
|
e
|
$-1$ |
The graph of is given above. The points and lie on the graph, as shown. What is the value of
The point on the graph has coordinates Therefore,
Applying to both sides of the above equation, and using the fact that we get
Therefore, we conclude that
The graph of $y = f(x)$ is given above. The points $A, B, C, D,$ and $E$ lie on the graph, as shown. What is the value of $f^{-1}(1)?$
|
a
|
$2$ |
|
b
|
$-2$ |
|
c
|
$-\dfrac 1 2$ |
|
d
|
$0$ |
|
e
|
$-1$ |
The graph of $y = f(x)$ is given above. The points $A,B,C,$ and $D$ lie on the graph, as shown. What is the value of $f^{-1}(1)?$
|
a
|
$3$ |
|
b
|
$-1$ |
|
c
|
$-\dfrac 1 3$ |
|
d
|
$2$ |
|
e
|
$\dfrac 1 3$ |
Recall that the "reverses" the action of so that which can be visualized using the mapping diagram, as shown below.
Since is just the composition we can now give the algebraic definition of the inverse.
The function is called the inverse of the function if for each in the domain of
In other words, to show that two functions and are inverses of each other, we need to check the following condition:
Suppose that and If is the inverse of then what is the value of ?
If is the inverse of then we must have
To find we write down the function and replace every occurrence of with
Then, we substitute into the right-hand side of our expression and simplify, as follows:
Because is the inverse of we also know that Equating the above expression with and solving for we get
Suppose that $f(x)=x-6$ and $g(x)=x + k.$ If $g$ is the inverse of $f,$ then $k=$
|
a
|
$-\dfrac{1}{6}$ |
|
b
|
$\dfrac{1}{6}$ |
|
c
|
$-6$ |
|
d
|
$1$ |
|
e
|
$6$ |
Suppose that $f(x)=2x$ and $g(x)=\dfrac x k.$ If $g$ is the inverse of $f,$ then $k=$
|
a
|
$-2$ |
|
b
|
$\dfrac{1}{2}$ |
|
c
|
$2$ |
|
d
|
$1$ |
|
e
|
$-\dfrac{1}{2}$ |
Suppose that $f(x)=\dfrac{x}{3}+7$ and $g(x)=3x+k.$ If $g$ is the inverse of $f,$ then $k=$
|
a
|
$-7$ |
|
b
|
$-21$ |
|
c
|
$21$ |
|
d
|
$7$ |
|
e
|
$3$ |