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Intro to inverse functions

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, start superscript, minus, 1, end superscript (and read as "f inverse"), will reverse this mapping. Function f, start superscript, minus, 1, end superscript 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:
Choose 1 answer:

Defining inverse functions

In general, if a function f takes a to b, then the inverse function, f, start superscript, minus, 1, end superscript, 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, left parenthesis, a, right parenthesis, equals, b, \Longleftrightarrow, f, start superscript, minus, 1, end superscript, left parenthesis, b, right parenthesis, equals, 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 mapping diagram above. What is h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis?

Solution

We are given information about function h and are asked a question about function h, start superscript, minus, 1, end superscript. Since inverse functions reverse each other, we need to reverse our thinking.
Specifically, to find h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, we can find the input of h whose output is 9. This is because if h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, equals, x, then by definition of inverses, h, left parenthesis, x, right parenthesis, equals, 9.
From the mapping diagram, we see that h, left parenthesis, 6, right parenthesis, equals, 9, and so h, start superscript, minus, 1, end superscript, left parenthesis, 9, right parenthesis, equals, 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, start superscript, minus, 1, end superscript, left parenthesis, 3, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Example 2: Graph

This is the graph of function g. Let's find g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis.
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, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, we can find the input of g that corresponds to an output of minus, 7. This is because if g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, equals, x, then by definition of inverses, g, left parenthesis, x, right parenthesis, equals, minus, 7.
From the graph, we see that g, left parenthesis, minus, 3, right parenthesis, equals, minus, 7.
Therefore, g, start superscript, minus, 1, end superscript, left parenthesis, minus, 7, right parenthesis, equals, minus, 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, start superscript, minus, 1, end superscript, left parenthesis, 4, right parenthesis?
Choose 1 answer:
Choose 1 answer:

Challenge problem
Given that f, left parenthesis, x, right parenthesis, equals, 3, x, minus, 2, what is f, start superscript, minus, 1, end superscript, left parenthesis, 7, right parenthesis?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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, left parenthesis, x, right parenthesis
minus, 2start fraction, 1, divided by, 4, end fraction
minus, 1start fraction, 1, divided by, 2, end fraction
01
12
24
We can reverse the inputs and outputs of function f to find the inputs and outputs of function f, start superscript, minus, 1, end superscript. So if left parenthesis, a, comma, b, right parenthesis is on the graph of y, equals, f, left parenthesis, x, right parenthesis, then left parenthesis, b, comma, a, right parenthesis will be on the graph of y, equals, f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis.
This gives us these graph and table of values of f, start superscript, minus, 1, end superscript.
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, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis
start fraction, 1, divided by, 4, end fractionminus, 2
start fraction, 1, divided by, 2, end fractionminus, 1
10
21
42
Looking at the graphs together, we see that the graph of y, equals, f, left parenthesis, x, right parenthesis and the graph of y, equals, f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis are reflections across the line y, equals, 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, equals, x.

Check your understanding

Problem 3
This is the graph of y, equals, h, left parenthesis, x, right parenthesis.
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, equals, h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis?
Choose 1 answer:
Choose 1 answer:

Problem 4
The graph of y, equals, h, left parenthesis, x, right parenthesis is a line segment joining the points left parenthesis, 5, comma, 1, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis.
Drag the endpoints of the solid segment below to graph y, equals, h, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis.

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, equals, start fraction, 5, divided by, 9, end fraction, left parenthesis, F, minus, 32, right parenthesis 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, equals, start fraction, 9, divided by, 5, end fraction, C, plus, 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.

Want to join the conversation?

  • duskpin ultimate style avatar for user Izzy
    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.
    (7 votes)
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    • leaf green style avatar for user kubleeka
      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.
      (3 votes)
  • orange juice squid orange style avatar for user Nathan Baker
    i have trouble understanding inverses. Can someone help me?
    i have trouble solving problems for the inverses.
    (2 votes)
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    • hopper cool style avatar for user obiwan kenobi
      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.
      (6 votes)
  • blobby green style avatar for user m.holder1324
    how would I find the inverse function of a quadratic, such as 2x^2+2x-1?
    (2 votes)
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    • starky ultimate style avatar for user Paul Miller
      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.
      (5 votes)
  • blobby green style avatar for user annaturner
    How did you get -3 for the second example? I see no correlation between the -7 and -3...
    (4 votes)
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  • blobby green style avatar for user ravzz96
    domain of f(x) is the range of inverse function and domain of inverse function is the range of f(x). but it is not true in some cases like f(x) = √2x-3. if we see domain of this function is x>=3/2 and inverse of this function is x^2/2+3/2 domain of this function is all real numbers . so, acc. to above line it says range of f(x) is all real numbers but in actual and it is x>=0.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      If f(x) = √(2x-3)
      Domain = x>= 3/2; Range = y >= 0

      Then for the inverse of f(x) = x^2/2+3/2
      Domain = x >= 0; Range = y >= 3/2

      If you widen the domain for the inverse function to x = any real number, then you will have input values for the inverse that can not be used in the original function. If you truly want the 2 functions to be inverses, you need to maintain the restrictions on domain/range for the 2 function.

      Hope this makes sense.
      (5 votes)
  • starky ultimate style avatar for user Kenna
    Is it true that when you solve for an inverse of a function, you do PEMDAS backwards?
    (2 votes)
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  • blobby green style avatar for user 黄 俊鹏
    What is the f inverse of f(x)=x^2?
    (1 vote)
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  • aqualine ultimate style avatar for user Parangat Aditya
    Is it necessary that the functions and their inverses are reflected along the line y = x?
    (2 votes)
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  • leaf orange style avatar for user ΜΑΘΘΑΙΟΣ
    could you in any way use these inverse functions to make a circle? I wouldnt think so, cause i cant find one, but can anyone else
    (2 votes)
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  • blobby green style avatar for user Fahad Akhtar
    is ono-to-one the only function that is invertible?
    (0 votes)
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