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Course: Algebra (all content) > Unit 7

Lesson 19: Introduction to inverses of functions (Algebra 2 level)
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Algebra (all content)
Functions
Introduction to inverses of functions (Algebra 2 level)
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Intro to inverse functions

Google Classroom
Learn what the inverse of a function is, and how to evaluate inverses of functions that are given in tables or graphs.
Inverse functions, in the most general sense, are functions that "reverse" each other.
For example, here we see that function f takes 1 to x, 2 to z, and 3 to y.
A mapping diagram. The map is titled f. The first oval contains the values one, two, and three. The second oval contains the values x, y, and z. There is an arrow starting at one and pointing to x. There is an arrow starting at two and pointing at z. There is an arrow starting at three and pointing at y.
The inverse of f, denoted f−1 (and read as "f inverse"), will reverse this mapping. Function f−1 takes x to 1, y to 3, and z to 2.
A mapping diagram. The map is titled f inverse. The first oval contains the values x, y, and z. The second oval contains one, two, and three. There is an arrow starting at x and pointing to one. There is an arrow starting at y and pointing to three. There is an arrow starting z and pointing to two.
Reflection question
Which of the following is a true statement?
Choose 1 answer:

Defining inverse functions

In general, if a function f takes a to b, then the inverse function, f−1, takes b to a.
The value a goes into function f and becomes value B which goes into f inverse and becomes value A.
From this, we have the formal definition of inverse functions:

f(a)=b⟺f−1(b)=a

Let's dig further into this definition by working through a couple of examples.

Example 1: Mapping diagram

A mapping diagram. The map is titled h. The first oval contains the values zero, four, six, and nine. The second oval contains the values three, seven, nine, and twelve. There is an arrow starting at zero and pointing to seven. There is an arrow starting at four and pointing at three. There is an arrow starting at six and pointing at nine. There is an arrow starting at nine and pointing at twelve.
Suppose function h is defined by the mapping diagram above. What is h−1(9)?

Solution

We are given information about function h and are asked a question about function h−1. Since inverse functions reverse each other, we need to reverse our thinking.
Specifically, to find h−1(9), we can find the input of h whose output is 9. This is because if h−1(9)=x, then by definition of inverses, h(x)=9.
From the mapping diagram, we see that h(6)=9, and so h−1(9)=6.

Check your understanding

A mapping diagram. The map is titled g. The first oval contains the values negative one, zero, three, and five. The second oval contains the values two, three, four, and eight. There is an arrow starting at negative one and pointing at three. There is an arrow starting at zero and pointing at four. There is an arrow starting at three and pointing at eight. There is an arrow starting at five and pointing at two
Problem 1
g−1(3)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Example 2: Graph

This is the graph of function g. Let's find g−1(−7).
A coordinate plane. The x-axis scales by zero point five, and the y-axis scales by one. The function y equals g of x is a continuous curve that starts at negative three, negative seven and increases slowly to the point negative one, negative five. Then the graph increases faster through the point zero, negative five point five and one, negative three point five. It continues to increase at a faster rate through the point two, two and the point three, ten.

Solution

To find g−1(−7), we can find the input of g that corresponds to an output of −7. This is because if g−1(−7)=x, then by definition of inverses, g(x)=−7.
From the graph, we see that g(−3)=−7.
Therefore, g−1(−7)=−3.
A coordinate plane. The x-axis scales by zero point five, and the y-axis scales by one. The function y equals g of x is a continuous curve that starts at negative three, negative seven and increases slowly to the point negative one, negative five. Then the graph increases faster through the point zero, negative five point five and one, negative three point five. It continues to increase at a faster rate through the point two, two and the point three, ten. There is a dashed vertical line at x equals negative three and a vertical dashed line at y equals negative seven. These lines intersect at the point negative three, negative seven, which is plotted is labeled.

Check your understanding

A coordinate plane. The x- and y- axes each scale by zero point five. The function y equals h of x is a straight that line goes through the point, negative two, four, the point zero, three, and the point two, two.
Problem 2
What is h−1(4)?
Choose 1 answer:

Challenge problem
Given that f(x)=3x−2, what is f−1(7)?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

A graphical connection

The examples above have shown us the algebraic connection between a function and its inverse, but there is also a graphical connection!
Consider function f, given in the graph and in a table of values.
A coordinate plane. The x- and y- axes each scale by one. The function y equals f of x is a nonlinear curve that goes through the following points: the point negative two, one-fourth, the point negative one, one-half, the point zero, one, the point one, two, and the point two, four.
xf(x)
−214
−112
01
12
24
We can reverse the inputs and outputs of function f to find the inputs and outputs of function f−1. So if (a,b) is on the graph of y=f(x), then (b,a) will be on the graph of y=f−1(x).
This gives us these graph and table of values of f−1.
A coordinate plane. The x- and y- axes each scale by one. The function y equals f inverse of x is a nonlinear curve that goes through the following points: the point one-fourth, negative two, the point one-half, negative one, the point one, zero, the point two, one, and the point four, two.
xf−1(x)
14−2
12−1
10
21
42
Looking at the graphs together, we see that the graph of y=f(x) and the graph of y=f−1(x) are reflections across the line y=x.
A coordinate plane. The x- and y-axes both scale by one. There is a curved lines representing the function y equals f of x. The line is the equation y equals two to the power of x. There is another curved line representing the function y equals f inverse of x. The second line is a reflection of the first curved line over the line y equals x.
This will be true in general; the graph of a function and its inverse are reflections over the line y=x.

Check your understanding

Problem 3
This is the graph of y=h(x).
A coordinate plane. The x- and y-axes both scale by one. There is a straight line representing the function y equals h of x. The line goes through the points zero, negative two and six, zero
Which is the best choice for the graph of y=h−1(x)?
Choose 1 answer:

Problem 4
The graph of y=h(x) is a line segment joining the points (5,1) and (2,7).
Drag the endpoints of the solid segment below to graph y=h−1(x).

Why study inverses?

It may seem arbitrary to be interested in inverse functions but in fact we use them all the time!
Consider that the equation C=59(F−32) can be used to convert the temperature in degrees Fahrenheit, F, to a temperature in degrees Celsius, C.
But suppose we wanted an equation that did the reverse – that converted a temperature in degrees Celsius to a temperature in degrees Fahrenheit. This describes the function F=95C+32, or the inverse function.
On a more basic level, we solve many equations in mathematics, by "isolating the variable". When we isolate the variable, we "undo" what is around it. In this way, we are using the idea of inverse functions to solve equations.

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  • blobby green style avatar for user m.holder1324
    Posted 8 years ago. Direct link to m.holder1324's post “how would I find the inve...”
    how would I find the inverse function of a quadratic, such as 2x^2+2x-1?
    (41 votes)
    Default Khan Academy avatar avatar for user
    • starky ultimate style avatar for user Paul Miller
      Posted 8 years ago. Direct link to Paul Miller's post “You can find the inverse ...”
      Incredible Answer
      Great Answer
      Good Answer
      You can find the inverse of any function y=f(x) by reflecting it across the line y=x. The quadratic you list is not one-to-one, so you will have to restrict the domain to make it invertible.
      Algebraically reflecting a graph across the line y=x is the same as switching the x and y variables and then resolving for y in terms of x.
      As you progress in your ability to find inverse functions you can see Sal solve for an inverse of a quadratic function here:
      https://www.khanacademy.org/math/algebra2/manipulating-functions/finding-inverse-functions/v/function-inverses-example-2
      But i highly recommend you make sure you can find the inverse of a linear function first before tackling quadratics and the associated domain restriction complications that they bring. If after working through that video and the subsequent examples, you would be better served posting your question there if you still aren't sure.
      (81 votes)
  • starky ultimate style avatar for user Kenna
    Posted 5 years ago. Direct link to Kenna's post “Is it true that when you ...”
    Is it true that when you solve for an inverse of a function, you do PEMDAS backwards?
    (19 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      Posted 5 years ago. Direct link to Ian Pulizzotto's post “Nice question! Yes you c...”
      Great Answer
      Good Answer
      Nice question!

      Yes you could think of it that way. If a function can be constructed by starting with x and performing a sequence of (reversible) operations, then its inverse can be constructed by starting with x and both reversing each operation and reversing the order of operations.

      Example: Suppose f(x) = 7(x - 5)^3. Note that f(x) is constructed by starting with x, subtracting 5, cubing, and then multiplying by 7.

      Then f^-1(x) is constructed by starting with x, dividing by 7, taking the cube root, and then adding 5.
      So f^-1(x) = cuberoot(x/7) + 5.
      (48 votes)
  • duskpin ultimate style avatar for user Izzy
    Posted 6 years ago. Direct link to Izzy's post “Why is the inverse always...”
    Why is the inverse always a reflection? Is it simply two lines that have the same set of reversed relationships, because plugging in the answer does not make a full restitution, instead it gives the same original value of x in a different line? Is there another reason for this? I am fascinated.
    (8 votes)
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    • leaf green style avatar for user kubleeka
      Posted 6 years ago. Direct link to kubleeka's post “We can think of a functio...”
      Good Answer
      We can think of a function as a collection of points in the plane. Each point has the form (x, y). If we consider the inverse function, it will contain each of these points, but with the coordinates switched.

      So if (a, b) is on our original function, then (b, a) is on the inverse. Let's look at how we get from (a, b) to (b, a). Draw a line segment between them.

      The slope of this line segment is then (b-a)/(a-b)=(-1)(b-a)/(b-a)= -1. That's interesting; if we have a point on a function and want to find the corresponding point on the inverse function, we slide along a line of slope -1. But how far do we slide?

      Let's find the midpoint of our line segment. In the x-direction, we go from a to b. So the midpoint has the x-coordinate (a+b)/2. In the y-direction, we go from b to a. So the midpoint has y-coordinate (b+a)/2. Same as the x-coordinate!
      So the midpoint of the segment must lie on the line y=x. Notice that y=x has a slope of 1, and our segment has a slope of -1. So the two are perpendicular.

      So what we've done to move from (a, b) to (b, a) is reflect over the line y=x.
      (18 votes)
  • starky sapling style avatar for user Izabella Miller
    Posted 7 months ago. Direct link to Izabella Miller's post “i dont understand”
    i dont understand
    (8 votes)
    Default Khan Academy avatar avatar for user
  • orange juice squid orange style avatar for user Nathan Baker
    Posted 4 years ago. Direct link to Nathan Baker's post “i have trouble understand...”
    i have trouble understanding inverses. Can someone help me?
    i have trouble solving problems for the inverses.
    (5 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user obiwan kenobi
      Posted 4 years ago. Direct link to obiwan kenobi's post “An inverse function essen...”
      Good Answer
      An inverse function essentially undoes the effects of the original function. If f(x) says to multiply by 2 and then add 1, then the inverse f(x) will say to subtract 1 and then divide by 2. If you want to think about this graphically, f(x) and its inverse function will be reflections across the line y = x.

      To find the inverse of a function you just have to switch the x and the y and then solve for y.

      For example, what is the inverse of y = 2x + 1?

      y = 2x + 1
      x = 2y + 1. (Switch the x and y)
      2y = x - 1
      y = (x-1)/2. And we're done.
      (11 votes)
  • blobby green style avatar for user SPmath23
    Posted 8 months ago. Direct link to SPmath23's post “What about 3D graphs...or...”
    What about 3D graphs...or complex planes? Do inverse functions math work or is it just vectors?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • duskpin ultimate style avatar for user Alex Scott
      Posted 8 months ago. Direct link to Alex Scott's post “Yes, inverse functions wo...”
      Good Answer
      Yes, inverse functions work in 3D graphs and complex planes, not just in vectors.
      In mathematics, a 3D graph is a graph that shows a three-dimensional representation of a function or a set of data points. It is represented by three axes: x, y, and z. The x and y-axes represent the horizontal and vertical dimensions, respectively, while the z-axis represents the depth or height dimension.
      A complex plane is a two-dimensional plane that represents complex numbers. It is represented by two axes: the real axis and the imaginary axis. The real axis represents the real part of the complex number, while the imaginary axis represents the imaginary part of the complex number.
      Inverse functions can be graphed in 3D graphs and complex planes, just like in two-dimensional graphs. The graph of the inverse function is obtained by reflecting the original graph across the line y = x. The inverse function is defined only if the original function is one-to-one, which means that each input has a unique output.
      Vectors are also used in 3D graphs, but they are not the only mathematical concept used. Vector functions are used to describe curves and surfaces in three-dimensional space.
      In summary, inverse functions work in 3D graphs and complex planes, and they are graphed by reflecting the original graph across the line y = x. Vectors are also used in 3D graphs, but they are not the only mathematical concept used.
      (10 votes)
  • blobby green style avatar for user vklizardman06
    Posted 7 months ago. Direct link to vklizardman06's post “What about when you have ...”
    What about when you have multiple outputs for the function how do you solve the inverse?
    (3 votes)
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    • male robot donald style avatar for user Venkata
      Posted 7 months ago. Direct link to Venkata's post “Good question. This actua...”
      Good question. This actually happens in the case of inverse trigonometric functions, where one input gives infinite outputs. In this case, we restrict the range of the functions so that only a set amount of outputs are possible. For example, sin^(-1)(x) will only output values between [-pi/2,pi/2].
      (8 votes)
  • blobby green style avatar for user branpfeil
    Posted 9 months ago. Direct link to branpfeil's post “why is this so hard”
    why is this so hard
    (5 votes)
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  • sneak peak green style avatar for user steviecute
    Posted 2 months ago. Direct link to steviecute's post “Regardless of what I try ...”
    Regardless of what I try I still can't seem to grasp the majority of the concept involving inverse functions, it seems like it's just passing right through me. Does anyone have any tips?
    (4 votes)
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  • blobby green style avatar for user annaturner
    Posted 5 years ago. Direct link to annaturner's post “How did you get -3 for th...”
    How did you get -3 for the second example? I see no correlation between the -7 and -3...
    (4 votes)
    Default Khan Academy avatar avatar for user