Finding Inverses of Functions

The inverse of a function reverses its $x$ - $y$ coordinates. If, for a function $f,$ $x$ is an input and $y$ is its corresponding output, for the inverse, $f^{- 1}, y$ is the input and $x$ would be the corresponding output. $f (x) = y \Leftrightarrow f^{- 1} (y) = x$

Some function families are inverses of each other. This is because some functions undo each other. For example,

x^{2}

and

\pm x

are inverses because radicals and exponents (with the same index) undo each other.

Determine the inverse of the function

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Exercise

Some of the coordinates of the function $g$ are shown in the table. Find $g^{- 1},$ then graph $g$ and $g^{- 1}$ on the same coordinate plane.

$x$	$g (x)$
$- 4$	$3$
$- 2$	$2$
$0$	$1$
$2$	$0$
$4$	$- 1$

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Solution

An inverse of a function reverses its $x$ - and $y$ -coordinates. When a function is expressed as a table of values, finding its inverse means switching the coordinates. For example, the point $(- 4, 3)$ on $g$ becomes $(3, - 4)$ on $g^{- 1} .$ The following table describes $g^{- 1} (x) .$

$x$	$g^{- 1} (x)$
$3$	$- 4$
$2$	$- 2$
$1$	$0$
$0$	$2$
$- 1$	$4$

We can graph both $g$ and $g^{- 1}$ by marking the points from both tables on the same coordinate plane.

Method

Finding the Inverse of a Function

Depending on how a function is presented, finding its inverse can be done in different ways. When a function's rule is given, finding the inverse algebraically is advantageous. Consider the function

f (x) = \frac{2 x - 1}{3} .

1

Replace

f (x)

with

y

To begin, since $f (x) = y$ describes the input-output relationship of the function, replace $f (x)$ with $y$ in the function rule. $f (x) = \frac{2 x - 1}{3} \Leftrightarrow y = \frac{2 x - 1}{3}$

2

Switch

x

and

y

Because the inverse of a function reverses $x$ and $y,$ the variables can be switched. Notice that every other piece in the function rule remains the same. $y = \frac{2 x - 1}{3} \Rightarrow x = \frac{2 y - 1}{3}$

3

Solve for

y

Solve the resulting equation from Step 2 for

y .

Here, this will involve using the inverse operations.

x = \frac{2 y - 1}{3}

MultEqn

LHS \cdot 3 = RHS \cdot 3

3 x = 2 y - 1

AddEqn

LHS + 1 = RHS + 1

3 x + 1 = 2 y

DivEqn

LHS / 2 = RHS / 2

\frac{3 x + 1}{2} = y

RearrangeEqnRearrange equation

y = \frac{3 x + 1}{2}

4

Replace

y

with

f^{- 1} (x)

Just as $f (x) = y$ shows the input-output relationship of $f,$ so does $f^{- 1} (x) = y .$ Thus, replacing $y$ with $f^{- 1} (x)$ gives the rule for the inverse of $f .$

$y = \frac{3 x + 1}{2} \Leftrightarrow f^{- 1} = \frac{3 x + 1}{2}$

Notice that in $f,$ the input is multiplied by $2,$ decreased by $1$ and divided by $3 .$ From the rule of $f^{- 1},$ it can be seen that $x$ undergoes the inverse of these operation in the reverse order. Specifically, $x$ is multiplied by $3,$ increased by $1,$ and divided by $2 .$

Determine the inverse of the quadratic function

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Exercise

Consider the quadratic function $f (x) = 3 x^{2} .$ Find its inverse function for when $x > 0 .$

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Solution

To find the inverse of

f (x),

we first need to replace

f (x)

with

y .

f (x) = 3 x^{2} \Rightarrow y = 3 x^{2}

The next step is to switch

x

and

y

in the function rule.

y = 3 x^{2} ⟶ switch x = 3 y^{2}

Now we need to solve for

y .

The resulting equation will be the inverse of the given function.

x = 3 y^{2}

DivEqn

LHS / 3 = RHS / 3

\frac{x}{3} = y^{2}

RearrangeEqnRearrange equation

y^{2} = \frac{x}{3}

SqrtEqn

LHS = RHS

y = \pm \frac{x}{3}

Switching

x

and

y

also switches the domain and the range. This means that the restriction on the domain of the function,

x > 0,

becomes a restriction on the range of the inverse function,

y > 0 .

Thus, there are no negative

y

-values.

y = \frac{x}{3}

Finally, to indicate that this is the inverse of

f (x),

we will replace

y

with

f^{- 1} (x) .

y = \frac{x}{3} \Rightarrow f^{- 1} (x) = \frac{x}{3}

Finding Inverses of Functions

Example

Determine the inverse of the function

Finding the Inverse of a Function

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2

3

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Example

Determine the inverse of the quadratic function