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Inverse Function Worksheets

Our compilation of printable inverse function worksheets should be an obvious destination, if practicing undoing functions or switching input and output values is on your mind. High school students can scroll through a bunch of tried and tested exercises like observing graphs and determining if they are functions, verifying if two functions are inverses of each other, finding inverses of functions with restricted domains graphically and algebraically, to mention just a few. Our free inverse function worksheets are definitely worth a try!

Printing Help - Please do not print inverse functions worksheets directly from the browser. Kindly download them and print.

Is it a Function? | Does it Have an Inverse?

A vertical line test is all it takes to determine if a graph represents a function. Establish if it has a one-to-one correspondence and passes the horizontal line test as well to figure out if it has an inverse function.

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Verifying if Two Functions Are Inverses | Graphically

High school learners examine the graphs and sketch the line y = x; if the two functions f(x) and g(x) are mirror images of each other or reflect each other with y = x as the line of symmetry, then they are inverses of each other.

Are f(x) and g(x) Inverses? | Algebraically

Verify algebraically if the functions f(x) and g(x) are inverses of each other in a two-step process. Plug the value of g(x) in every instance of x in f(x), followed by substituting f(x) in g(x). Simplify and check if both result in x.

Finding the Inverse | Level 1

Equate f(x) with y. Swap x with y in each of the linear functions presented in these printable inverse function worksheets and solve for y, and replace y with the inverse f -1 (x) and check.

Finding the Inverse | Level 2

Nudge your high school learners into a whole new arena of finding the inverse of a function by challenging them with a variety of functions like rational functions, radical functions and logarithmic functions to revisit concepts.

Evaluating Inverse Functions | Graph

Get ready for spades of practice with these inverse function worksheet pdfs. Observe the graph keenly, where the given output or inverse f -1 (x) are the y-coordinates, and find the corresponding input values.

Evaluating Inverse Functions

Restricted domains being the primary focus of this batch of inverse function worksheet pdfs, instruct students to find f(x) or f -1 (x) by observing the values given as relation mapping diagrams or as tables.

Related Worksheets

» Identifying Functions

» Domain and Range

» Function Table

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Inverse Functions Worksheet

Students will practice work with inverse functions including identifying the inverse functions , graphing inverses and more.

Example Questions

Directions: Find the inverse of each function

Challenge Problems. Part II

• Inverse Function
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Inverse Functions (Worksheet with Solutions)

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

10 November 2019

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This worksheet ( with solutions ) helps students take the first steps in their understanding and in developing their skills and knowledge of finding the Inverse of a Function . Questions are carefully planned so that understanding can be developed, misconceptions can be identified and so that there is progression both across and down the sheet. An interactive version of this sheet is available at https://www.maths4everyone.com/skills/finding-the-inverse-by-working-backwards-2832.html

The interactive version allows individual questions to be selected for enlarged display onto a screen. The answer can then be worked out ‘live’ by the teacher (or student) or a single click will reveal my solution. This not only helps in class, but it is also very useful for a student who is working at home.

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Graphing Inverse Functions Worksheets

Have you noticed in almost all forms of mathematics that there are some operations that are the exact counter or mathematical kryptonite to one another? Addition cancel be countered by subtraction and multiplication by division. When it comes to functions the same rings true. If you have a function and want to counter its value, you would combine it with its natural inverse. Inverses are kinda the undo button for functions. If you know that f(x) created y, then if you took y and put it into the inverse of f, you would get the output of x. If you have the graph of function, the inverse of that graph is simply a mirrored reflect of itself across the origin of the graph. This selection of worksheets and lessons show students how to visualize the inverse of a function on a graph.

Aligned Standard: HSF-BF.B.4c

• Inverse Graphs Step-by-step Lesson - I walk you through two different methods. Some teachers like to use trig to solve this too.
• Guided Lesson - I give you two standing graphs that you can work off of here.
• Guided Lesson Explanation - Solve for x and then walk it through from there.
• Practice Worksheet - I made these purposely easy early on. This will help students build a little confidence. It does get more difficult as they proceed.
• Matching Worksheet - Match the graphs to the inverses of what is being presented.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

The inverse of square roots really scare students when we start this skill. Work through a problem with them.

• Homework 1 - We look at all methods of attack as an option.
• Homework 2 - Domain: x ≥ 10/9. Start by determining standard x and y values by plugging x values into the given function.
• Homework 3 - We would follow this through all points and point the lines. The blue line would result.

Practice Worksheets

The inverse really is just the function with an adjusted orientation.

• Practice 1 - A good hint is to identify the domain of f(x).
• Practice 2 - We would follow this through all points and point the lines. The blue line would result.
• Practice 3 - f of x (f(x)) is the square root of x + 11x +12 is what we are working with here.

Math Skill Quizzes

These graphs should print well in grayscale.

• Quiz 1 - Method 1: You could first find the algebraic inverse of the function and then just plot points.
• Quiz 2 - Method 2: You could pick points for x and determine y. Then take the inverse of those points (switch the x and y values). This is the method that most people use.
• Quiz 3 - This can be done a number of ways.

How to Graph the Inverse of Functions

Graphing functions is very easy. In case of linear functions, you have to find the y- and x-intercept and join the line but things get tricky when we are asked to graph the inverse of functions. The inverse of any function, on graph, is just a mirror image of it reflected over the line y = x. The line that is used for a guide to reflect it over the axis passes through the origin and has a slope of 1. There are three different methods that you can use to graph these inverses. We will start with a lengthy one and introduce you to the easiest one at the end.

Method 1: Find the Inverse of a Function - When you are asked to graph the inverse of a function you can start by finding the inverse of the function. To find the inverse of a function, you replace f(x) with y and then make x the subject of the equation. Once you make x the subject, you replace x with f(x) and replace y with x. You then take a set of points and plot them on the graph. It is the most time-consuming method. This is because you need to plot a whole bunch of ordered pairs, after you first calculate them.

Method 2: A Mirror Reflection - To make things easy for you, we will disclose a secret that will save your time. When you plot a function on the graph you can easily plot its inverse without finding the inverse function. An inverse of a function is reflection of the actual function over the line y = x. It is a line that passes through the origin and has a slope of 1. You can count the squares and reflect the function over this line.

Method 3: Swapping Places of Entire Domain and Range - The easiest and the quickest method to graph an inverse of a function is to interchange the values. How? - If a function passes through the following points; (-4, -11), (-2, -7), and (0, -3), for the inverse you will have to plot the following points; (-11, -4), (-7, -2), and (-3,0).

Where Does This Apply in The Real World?

Understand how to data flows through a function is very important in many aspects of technology. Many different forms of digital communication hinge on being able to breakdown the flow of data it uses. When your wireless phone or Wi-Fi router is having communication issues engineers will trouble shoot this by sending bits of wireless packets across the channel of this device from where it originates to the device itself. Engineers can see if the packet that they initiated was distorted or changed in anyway. The calculate this by using this skill to diagnose the exact issue. Eventually they will be able to pinpoint was is jamming or changing the quality of the data stream. This skill is also very helpful, in a similar fashion, to encrypted communication. Needless to say, this is a significant application in computer security. Researchers will often be able to pinpoint vulnerabilities or errors in programming code through this technique. If you plan to work in field that involves digital communication expect to use this skill regularly.

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6.1e: Exercises - Inverse Trigonometric Functions

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• Page ID 72369

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A: Concepts.

Exercise \(\PageIndex{A}\)

• Why do the functions \(f(x)=\sin^{-1}x\) and \(g(x)=\cos^{-1}x\) have different ranges?
• Since the functions \(y=\cos x\) and \(y=\cos^{-1}x\) are inverse functions, why is \(\cos^{-1}\left (\cos \left (-\dfrac{\pi }{6} \right ) \right )\)   not equal to \(-\dfrac{\pi }{6}\)?
• Explain the meaning of \(\dfrac{\pi }{6}=\arcsin (0.5)\).
• Most calculators do not have a key to evaluate \(\sec ^{-1}(2)\) .   Explain how this can be done using the cosine function or the inverse cosine function.
• Why must the domain of the sine function, \(\sin x\) ,   be restricted to \(\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]\) for the inverse sine function to exist?
• Discuss why this statement is incorrect: \(\arccos(\cos x)=x\) for all \(x\).
• Determine whether the following statement is true or false and explain your answer: \(\arccos(-x)=\pi - \arccos x\)

1. The function \(y=\sin x\) is one-to-one on \(\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]\) ;   thus, this interval is the range of the inverse function of \(y=\sin x\), \(f(x)=\sin^{-1}x\)   The function \(y=\cos x\) is one-to-one on \([0,\pi ]\) ;   thus, this interval is the range of the inverse function of \(y=\cos x\), \(f(x)=\cos^{-1}x\)

3. \(\dfrac{\pi }{6}\) is the radian measure of an angle between \(-\dfrac{\pi }{2}\) and \(\dfrac{\pi }{2}\) whose sine is \(0.5\).

5. In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval \(\left [ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right ]\) so that it is one-to-one and possesses an inverse.

7. True . The angle, \(\theta _1\) that equals \(\arccos(-x)\), \(x>0\), will be a second quadrant angle with reference angle, \(\theta _2\), where \(\theta _2\) equals \(\arccos x\), \(x>0\). Since \(\theta _2\) is the reference angle for \(\theta _1\), \(\theta _2=\pi - \theta _1\) and \(\arccos(-x)=\pi - \arccos x-\)add texts here. Do not delete this text first.

B: Evaluate Inverse Trigonometric Functions for "Special Angles"

Exercise \(\PageIndex{B}\)

\( \bigstar \)  Evaluate the expressions.

9.  \( 0\)     11.  \( 0\)     13.  \( 0\)     15.  \(\dfrac{\pi }{6}\)     17.  \(\dfrac{\pi }{6}\)     19.  \(\dfrac{\pi }{4}\)     21.  \(\dfrac{\pi }{6}\)      23.  \(-\dfrac{\pi }{2}\)     25.  \(-\dfrac{\pi }{4}\)    27.  \(\dfrac{2\pi }{3}\)     29.  \(\dfrac{3\pi }{4}\)     31.  \( - \dfrac{\pi }{3}\)    33.  \(-\dfrac{\pi }{3}\)     35.  \( \tan^{-1} \left(\dfrac{-1}{\sqrt{3}}\right) = - \dfrac{\pi}{6}\)  

C: Evaluate Inverse Trigonometric Functions with a Calculator

Exercise \(\PageIndex{C}\)

\( \bigstar \)  Use a calculator to evaluate each expression. Express answers to the nearest hundredth.

37.  \(1.98\)      39.  \(1.41\)      41.   \(0.73\)

D: Evaluate \( f^{-1} (f( \theta )) \) Compositions

Exercise \(\PageIndex{D}\)

\( \bigstar \)  Evaluate without a calculator.

45.  \(  \dfrac{\pi}{10}   \)       47.  \( - \dfrac{\pi}{4}   \)       49.  \(  \dfrac{\pi}{2}  \)       51.  \(  \dfrac{\pi}{10}   \)       53.  \(  \dfrac{2\pi}{3}  \)       55.  \(  \dfrac{4\pi}{5}   \)       57.  \( - \dfrac{\pi}{3}   \)       59.  \( - \dfrac{\pi}{9}   \)       61.  \( - \dfrac{\pi}{8}   \)

E: Evaluate \( f (f^{-1}( \frac{a}{b} )) \) Compositions

Exercise \(\PageIndex{E}\)

65.   \(\dfrac{1}{5}\)  \(\qquad\)  67.   \( 4 \)  \(\qquad\)  69.   undefined  

F: Evaluate \( f (g^{-1}( \frac{a}{b} )) \) Compositions

Exercise \(\PageIndex{F}\)

71.  \(\dfrac{\sqrt{5}}{5}\)            73.  \(\dfrac{4}{5}\)       75.  \(- \dfrac{2}{3}\)      77.  \(\dfrac{\sqrt{3}}{2}\)         79.  \(\dfrac{5}{13}\)  81.  \( - 2 \)           83.  \( - \dfrac{\sqrt{3}}{3}\)            85.  \( \dfrac{\sqrt{3}}{3}\)            87.  \(\dfrac{4\sqrt{7}}{7}\)       89.  \(\dfrac{3\sqrt{7}}{7}\)   

G: Evaluate \( f^{-1}(g( \theta )) \) Compositions

Exercise \(\PageIndex{G}\)

\( \bigstar \)  Find the angle \(\theta\) in the given right triangle. Round answers to the nearest hundredth.

\( \bigstar \)  Find the exact value, if possible, without a calculator, or round to the nearest hundredth. 

103.  \(0.56\)  radians          105.  \(0\)        107. undefined            109. \(  \dfrac{3\pi}{4}  \)          111.  \(0\)         113.  \(0.71\)      115.  \(-\dfrac{\pi}{4}\)       117. \(  0.86   \)  

H: Evaluate  \( f (g^{-1}( h(u) )) \) Compositions

Exercise \(\PageIndex{H}\)

For the exercises below, (a) Find the exact value of the expression in terms of \(u\). (b) State any restrictions to \(u\).

121. \( \sqrt{1-u^2} \), \( -1 \le u \le 1 \)        123. \( \dfrac{u}{\sqrt{1+u^2} } \), no restrictions        125. \( \dfrac{\sqrt{1-u^2}}{u} \), \( -1 \le u \le 1 \)      127. \(\dfrac{u-1}{\sqrt{-u^2+2u}}\), \( 0 \le u \le 2 \)        129. \(\dfrac{\sqrt{u^2-1}}{u}\), \( u \ge 1 \) or \( u \le -1 \)        131. \(\dfrac{u+\tfrac{1}{2}}{\sqrt{u^2+u+\tfrac{5}{4}}}\), no restrictions  133. \(\dfrac{u}{u+1}\), \( u \gt -\dfrac{1}{2} \)      

I: Graphs of inverses

Exercise \(\PageIndex{I}\)

138. Graph \(y=\sin^{-1} x\) and state the domain and range of the function.

139. Graph \(y=\arccos x\)   and state the domain and range of the function.

140. Graph one cycle of \(y=\tan^{-1} x\) and state the domain and range of the function.

J: Applications

Exercise \(\PageIndex{J}\)

143. Suppose a \(13\)-foot ladder is leaning against a building, reaching to the bottom of a second-floor window \(12\) feet above the ground. What angle, in radians, does the ladder make with the building?

144. Suppose you drive \(0.6\) miles on a road so that the vertical distance changes from \(0\) to \(150\) feet. What is the angle of elevation of the road?

145. An isosceles triangle has two congruent sides of length \(9\) inches. The remaining side has a length of \(8\) inches. Find the angle that a side of \(9\) inches makes with the \(8\)-inch side.

146. Without using a calculator, approximate the value of \(\arctan (10,000)\) .   Explain why your answer is reasonable.

147. A truss for the roof of a house is constructed from two identical right triangles. Each has a base of \(12\) feet and height of \(4\) feet. Find the measure of the acute angle adjacent to the \(4\)-foot side.

148. The line \(y=\dfrac{3}{5}x\) passes through the origin in the \(x,y\)-plane. What is the measure of the angle that the line makes with the positive \(x\)-axis?

149. The line \(y=\dfrac{-3}{7}x\) passes through the origin in the \(x,y\) -plane. What is the measure of the angle that the line makes with the negative \(x\)-axis?

150. What percentage grade should a road have if the angle of elevation of the road is \(4\) degrees? (The percentage grade is defined as the change in the altitude of the road over a \(100\)-foot horizontal distance. For example a \(5\%\) grade means that the road rises \(5\) feet for every \(100\) feet of horizontal distance.)

151. A \(20\)-foot ladder leans up against the side of a building so that the foot of the ladder is \(10\) feet from the base of the building. If specifications call for the ladder's angle of elevation to be between \(35\) and \(45\) degrees, does the placement of this ladder satisfy safety specifications?

152. Suppose a \(15\)-foot ladder leans against the side of a house so that the angle of elevation of the ladder is \(42\) degrees. How far is the foot of the ladder from the side of the house?

143.  \(0.395\)  radians         145.  \(1.11\)  radians      147.  \(1.25\)  radians      149.  \(0.405\)  radians    151.  No. The angle the ladder makes with the horizontal is \(60\) degrees..

1-7 Inverse Relations and Functions

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Inverse relations and functions worksheet, word docs, & powerpoints.

1-7 Assignment - Inverse Relations and Functions 1-7 Bell Work - Inverse Relations and Functions 1-7 Exit Quiz -Inverse Relations and Functions 1-7 Guided Notes SE - Inverse Relations and Functions 1-7 Guided Notes TE - Inverse Relations and Functions 1-7 Lesson Plan - Inverse Relations and Functions 1-7 Online Activities - Inverse Relations and Functions 1-7 Slide Show - Inverse Relations and Functions

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