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Given a function $f,$ sometimes it is possible to find another function that *undoes* $f.$ This particular function is called the *inverse function of* $f.$ This lesson will discuss how to verify whether two functions are inverse and will explain the procedure to find the inverse of a linear function.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Consider a function $f$ that takes any real number and returns a number greater by two units.
*undoes* $f?$

What would be a function that

Discussion

An inverse relation is a relation that interchanges the input and output values of a specific relation. In other words, it is the relation that *undoes* a certain relation. Consider the mapping diagram that shows a relation $p$ that takes elements from set $A$ and returns elements from set $B.$ *undoes* $p$ is $q.$ In this case, $q$ takes elements from set $B$ and returns elements from set $A.$ Relations can also be represented by a collection of ordered pairs. Here, the inverse of a relation is obtained by interchanging the elements of the ordered pairs of the original relation.

The relation that

$Relationp{(-1,C),(0,E),(5,Z),(10,W),(20,Q)}Inverse Relationq{(C,-1),(E,0),(Z,5),(W,10),(Q,20)} $

Graphically, the inverse of a relation is a reflection across the line $y=x.$
If the inverse relation of a function is also a function, then it is called an inverse function.

Discussion

Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function $f$ is another function $f_{-1}$ such that they *undo* each other.

$f(f_{-1}(x))=xandf_{-1}(f(x))=x$

Also, if $x$ is the input of a function $f$ and $y$ its corresponding output, then $y$ is the input of $f_{-1}$ and $x$ its corresponding output.

$f(x)=y⇔f_{-1}(y)=x$

$f(x)=2x−3andf_{-1}(x)=2x+3 $

These functions will be shown to $f(f_{-1}(x))=?x⇕2f_{-1}(x)−3=?x $

Now, in the above equation, $2x+3 $ will be substituted for $f_{-1}(x).$
$2f_{-1}(x)−3=?x$

Substitute

$f_{-1}(x)=2x+3 $

$2(2x+3 )−3=?x$

▼

Simplify left-hand side

DenomMultFracToNumber

$2⋅2a =a$

$(x+3)−3=?x$

RemovePar

Remove parentheses

$x+3−3=?x$

SubTerm

Subtract term

$x=x✓$

Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|

$f(f_{-1}(x))=?x$ | $2f_{-1}(x)−3=?x$ | $2(2x+3 )−3=?x$ | $x=x✓$ |

$f_{-1}(f(x))=?x$ | $2f(x)+3 =?x$ | $22x−3+3 =?x$ | $x=x✓$ |

Therefore, $f$ and $f_{-1}$ *undo* each other. The graphs of these functions are each other's reflection across the line $y=x.$ This means that the points on the graph of $f_{-1}$ are the *reversed* points on the graph of $f.$

Example

Kriz enjoys playing video games with their friends. For their birthday they received a copy of Mathleaks: The Adventure,

a math-based video game.

$f(x)=4x+5andg(x)=41 x−45 $

Help Kriz beat the level and get one step closer beating the game by mastering math!
**Graph:**

**Are $f$ and $g$ Inverse Functions?** Yes.

Graph both functions on the same coordinate plane and see if they are each other's reflection across the line $y=x.$

Since $f$ and $g$ are linear functions written in slope-intercept form, they can both be graphed using their slope and $y-$intercept.

Now, it can be seen whether the lines are each other's reflection across $y=x.$
The lines are indeed each other's reflection across $y=x.$ Therefore, $f$ and $g$ are inverse functions.

Pop Quiz

It was discussed previously that if the graphs of two functions are each other's reflection across $y=x,$ then they are inverse functions. Therefore, it can be determined if two functions are inverse just by looking at their graphs. Determine whether the following linear functions are inverse by looking at the lines.

Example

If Paulina gets an A on a math test, her mother will buy her a new saxophone.

To get an A, Paulina must answer two questions correctly. Help her get the new saxophone!

a Given $f(x)=5x+10$ and $g(x)=51 x−2,$ calculate $f(g(x))$ and $g(f(x)),$ and determine whether $f$ and $g$ are inverse functions.

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["x"]}}

{"type":"choice","form":{"alts":["Yes","No"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}

b Given $h(x)=21 x+6$ and $k(x)=2x−6,$ calculate $h(k(x))$ and $k(h(x)),$ and determine whether $h$ and $k$ are inverse functions.

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{"type":"choice","form":{"alts":["Yes","No"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

a If $f(g(x))=x$ and $g(f(x))=x,$ then $f$ and $g$ are inverse functions.

b If $h(k(x))=x$ and $k(h(x))=x,$ then $h$ and $k$ are inverse functions.

a Consider the given functions.

$f(x)=5x+10andg(x)=51 x−2 $

The composite function $f(g(x))$ will be found first. To do so, the definition of $f$ will be used.
$f(g(x))=5g(x)+10 $

Next, $g(x)=51 x−2$ can be substituted in the above equation.
$f(g(x))=5g(x)+10$

Substitute

$g(x)=51 x−2$

$f(g(x))=5(51 x−2)+10$

▼

Simplify right-hand side

Distr

Distribute $5$

$f(g(x))=5(51 x)−5(2)+10$

AssociativePropMult

Associative Property of Multiplication

$f(g(x))=5(51 )x−5(2)+10$

DenomMultFracToNumber

$5⋅5a =a$

$f(g(x))=1x−5(2)+10$

Multiply

Multiply

$f(g(x))=x−10+10$

AddTerms

Add terms

$f(g(x))=x$

Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|

$f(g(x))$ | $f(g(x))=5g(x)+10$ | $f(g(x))=5(51 x−2)+10$ | $f(g(x))=x$ |

$g(f(x))$ | $g(f(x))=51 f(x)−2$ | $g(f(x))=51 (5x+10)−2$ | $g(f(x))=x$ |

It has been found that both $f(g(x))$ and $g(f(x))$ are equal to $x.$ Therefore, $f$ and $g$ are inverse functions.

b Consider the given functions.

$h(x)=21 x+6andk(x)=2x−6 $

By following the same procedure as in Part A, the expressions for $h(k(x))$ and $k(h(x))$ will be found. Definition of First Function | Substitute Second Function | Simplify | |
---|---|---|---|

$h(k(x))$ | $h(k(x))=21 k(x)+6$ | $h(k(x))=21 (2x−6)+6$ | $h(k(x))=x+3$ |

$k(h(x))$ | $k(h(x))=2h(x)−6$ | $k(h(x))=2(21 x+6)−6$ | $k(h(x))=x+6$ |

It has been found that neither $h(k(x))$ nor $k(h(x))$ is equal to $x.$ Therefore, $h$ and $k$ are **not** inverse functions.

Pop Quiz

For the functions $f$ and $g,$ use a composition to determine whether they are inverse functions.

Discussion

A function can be represented by a table of values, a graph, a mapping diagram, or a function rule, among other ways. Depending on how the function is presented, finding its inverse can be done in different ways. When a function rule is given, finding the inverse algebraically is advantageous. Consider the following example function.
*expand_more*
*expand_more*
*expand_more*
*expand_more*

$f(x)=32x−1 $

There is a series of steps to follow in order to find the inverse function $f_{-1}(x).$
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)=32x−1 →y=32x−1 $

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=32x−1 switch x=32y−1 $

3

Solve for $y$

Solve the resulting equation from the previous step for $y.$ This will involve using the inverse operations.

$x=32y−1 $

▼

Solve for $y$

MultEqn

$LHS⋅3=RHS⋅3$

$3x=2y−1$

AddEqn

$LHS+1=RHS+1$

$3x+1=2y$

DivEqn

$LHS/2=RHS/2$

$23x+1 =y$

RearrangeEqn

Rearrange equation

$y=23x+1 $

4

Replace $y$ With $f_{-1}(x)$

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

$y=23x+1 →f_{-1}(x)=23x+1 $

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 operations in the reverse order. Specifically, $x$ is multiplied by $3,$ increased by $1,$ and divided by $2.$ Example

Zosia lives in Honolulu, Hawaii.

She is obsessed with math. While watching a surfer catch some gnarly waves, she came up with a function that models the distance of the surfer from the shore.$f(x)=-34 x+24 $

Here, $f(x)$ is the distance of the surfer from the shore after $x$ seconds of riding a wave. Given the distance from shore, Zosia now wants to finds the function that gives the time, in seconds, the surfer has been riding a wave. To do so, she needs to find $f_{-1},$ the inverse of $f.$ Help Zosia find this function! Write the answer in slope-intercept form. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">-<\/span><\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-\\dfrac{3}{4}x+18","-0.75x+18","\\dfrac{-3}{4}x+18","\\dfrac{3}{-4}x+18"]}}

Start by replacing $f(x)$ with $y.$ After that, switch the $x-$ and $y-$variables. Then, solve the obtained equation for $y.$ Finally, replace $y$ with $f_{-1}(x).$

To find the inverse of the function, there are four steps to follow.

- Replace $f(x)$ with $y.$
- Switch $x$ and $y.$
- Solve the obtained equation for $y.$
- Replace $y$ with $f_{-1}(x).$

These steps will be done one at a time.

$f(x)=-34 x+24→y=-34 x+24 $

$y=-34 x+24switch x=-34 y+24 $

$x=-34 y+24$

▼

Solve for $y$

SubEqn

$LHS−24=RHS−24$

$x−24=-34 y$

MultEqn

$LHS⋅(-43 )=RHS⋅(-43 )$

$(x−24)(-43 )=y$

Distr

Distribute $(-43 )$

$x(-43 )−24(-43 )=y$

CommutativePropMult

Commutative Property of Multiplication

$-43 x−24(-43 )=y$

MultPosNeg

$a(-b)=-a⋅b$

$-43 x−(-24(43 ))=y$

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$-43 x−(-472 )=y$

CalcQuot

Calculate quotient

$-43 x−(-18)=y$

SubNeg

$a−(-b)=a+b$

$-43 x+18=y$

RearrangeEqn

Rearrange equation

$y=-43 x+18$

$y=-43 x+18↓f_{-1}(x)=-43 x+18 $

Pop Quiz

Find the inverse of the given function. Write the answer in slope-intercept form — in the format $y=mx+b.$ If the slope and $y-$intercept are not integers, express them as decimal numbers. If necessary, round them to two decimal places.

Closure

In this lesson, it has been seen that the function that *undoes* a function $f$ is its inverse $f_{-1}.$ Consider the function given in the challenge presented at the beginning of the lesson.
### Hint

### Solution

### Step $1$

Here, the variable $y$ will be substituted for $f(x)$ in the given function.
### Step $2$

In this step, the variables $x$ and $y$ must be switched.
### Step $3$

Here, the equation obtained in the previous step will be solved for $y.$
### Step $4$

Finally, $f_{-1}(x)$ will be substituted for the $y-$variable in the equation obtained in the previous step.
*undoes* it, is $f_{-1}(x)=x−2$

Find the inverse function of $f(x)=x+2.$

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Start by replacing $f(x)$ with $y.$ After that, switch the $x-$ and $y-$variables. Then, solve the obtained equation for $y,$ and finally replace $y$ with $f_{-1}(x).$

To find the inverse of the function, there are four steps to follow.

- Replace $f(x)$ with $y.$
- Switch $x$ and $y.$
- Solve the obtained equation for $y.$
- Replace $y$ with $f_{-1}(x).$

These steps will be done one at a time.

$f(x)=x+2substitute y=x+2 $

$y=x+2switch x=y+2 $

$x=y+2⇔y=x−2 $

$y=x−2substitute f_{-1}(x)=x−2 $

It has been found that the inverse function of $f(x)=x+2,$ and therefore the function that