Explainer: Inverse of a Function

In this explainer, we will learn how to find the inverse of a function by changing the subject of the formula.

The concept of the inverse of a function builds on the concept of a function. First, make sure you are familiar with the key vocabulary we are going to use.

Definition: Key Vocabulary

A function associates an output called 𝑓(π‘₯) to an input π‘₯. It is defined by an equation that shows how the output is calculated given the input. For instance, the function 𝑓(π‘₯)=12π‘₯+3 associates to any π‘₯ half of its value plus 3.

If we plot the graph 𝑦=𝑓(π‘₯), we get a line that represents the mapping of all possible input values (π‘₯) onto their corresponding output values (𝑦). This means that any point of coordinates (π‘₯,𝑦) on this line gives a pair of solutions to the equation 𝑦=12π‘₯+3.

The graph can then be used to find an output value 𝑦 given an input value π‘₯.

The domain of a function is the set of all possible input numbers (on the π‘₯-axis). The range of a function is the set of all outputs (on the 𝑦-axis).

Now that the concept of function is clear in your mind, we can define the inverse of a function.

Definition: Inverse of a Function

The inverse of a function is a function that β€œgoes backwards”: if 𝑓 applied to an input π‘₯ gives an output 𝑦, then the inverse of 𝑓 (called π‘“οŠ±οŠ§) applied to 𝑦 gives π‘₯: 𝑓(π‘₯)=𝑦 and 𝑓(𝑦)=π‘₯.

Example 1: Determining Whether Two Functions Are Inverses of One Another

Let 𝑓(π‘₯)=3π‘₯+5 and 𝑔(π‘₯)=π‘₯βˆ’53. Is it true that 𝑓 is the inverse of 𝑔 and 𝑔 is the inverse of 𝑓?

Answer

We could try and find equations for inverses, but we should remember that the composition of a function with its inverse is the identity (π‘“βˆ˜π‘”)(π‘₯)=𝑓(𝑔(π‘₯))=𝑓π‘₯βˆ’53=3ο€Όπ‘₯βˆ’53+5=(π‘₯βˆ’5)+5=π‘₯, while (π‘”βˆ˜π‘“)(π‘₯)=𝑔(𝑓(π‘₯))=𝑔(3π‘₯+5)=(3π‘₯+5)βˆ’53=3π‘₯3=π‘₯.

We conclude that 𝑓 and 𝑔 are inverses of each other.

The first way to find the inverse of a function is to find its graph from the graph of the original function.

How to: Plot the Inverse of a Function from Its Graph

From the definition of the inverse of a function, we see that, for the inverse function, the roles of π‘₯ and 𝑦 are swapped over with respect to the original function. It means that if we flip the whole graph so that the 𝑦-axis is in the previous position of the π‘₯-axis and vice-versa, then we get the graph of the inverse function. This geometric transformation is actually a reflection in the line 𝑦=π‘₯.

For instance, the diagram below shows 𝑦=𝑓(π‘₯)=12π‘₯+3 (black line) and 𝑦=𝑓(π‘₯) (red line).

We can easily check with the two graphs that for any π‘₯; if 𝑓(π‘₯)=𝑦, then 𝑓(𝑦)=π‘₯. For instance, with π‘₯=2, 𝑓(2)=4 and 𝑓(4)=2.

When finding the inverse π‘“οŠ±οŠ§ of the function 𝑓, 𝑦 becomes the input and π‘₯ the output. As we have just seen, this process is represented graphically as a reflection in the line 𝑦=π‘₯. What about the domain and the range of the inverse of the function 𝑓?

About: Domain and Range of the Inverse of a Function

By indicating the domain and range of a function 𝑓 on the π‘₯-axis and 𝑦-axis, one can visualize that the domain and range of 𝑓 are swapped for π‘“οŠ±οŠ§: the range of 𝑓 is the domain of π‘“οŠ±οŠ§ and the domain of 𝑓 is the range of π‘“οŠ±οŠ§.

The second way to find the function π‘“οŠ±οŠ§, the inverse of function 𝑓, is to start from the equation of 𝑓 and use algebra to find the equation of π‘“οŠ±οŠ§.

How to: Find the Inverse of a Function Algebraically

We know that the function π‘“οŠ±οŠ§ is the inverse of function 𝑓 if for all points (π‘₯,𝑦) that satisfy 𝑓(π‘₯)=𝑦 we have 𝑓(𝑦)=π‘₯. We want now to find an expression where π‘₯ is the input and 𝑓(π‘₯)=π‘¦οŠ±οŠ§ is the output. With respect to 𝑓(π‘₯)=𝑦, this means that π‘₯ and 𝑦 are swapped.

Starting from 𝑓(π‘₯)=𝑦, we substitute π‘₯ with 𝑦 and 𝑦 with π‘₯. Then, we rearrange the equation in the form 𝑦=𝑓(π‘₯).

Finally, the range of 𝑓 needs to be determined in order to state the domain of π‘“οŠ±οŠ§.

Example 2: Finding the Inverse of a Function Algebraically

Find 𝑓(π‘₯) for 𝑓(π‘₯)=12π‘₯+3.

Answer

  1. We swap π‘₯ for 𝑦 and 𝑦 for π‘₯ in 12π‘₯+3=𝑦. This gives 12𝑦+3=π‘₯.
  2. We rearrange the previous equation in the form 𝑦=𝑓(π‘₯).
    1. We take away 3 from each side of the equation: 12𝑦=π‘₯βˆ’3.
    2. We multiply each side by 2: 𝑦=2(π‘₯βˆ’3).
    3. We expand the brackets on the right-hand side of the equation: 𝑦=2π‘₯βˆ’6.
  3. We find the range of 𝑓: all the real numbers. The domain does not need to be specified in this case.
  4. We write our solution: 𝑓(π‘₯)=2π‘₯βˆ’6.

Example 3: Finding the Inverse of a Function Algebraically

Find 𝑓(π‘₯) for 𝑓(π‘₯)=3π‘₯βˆ’2.

Answer

  1. We swap π‘₯ for 𝑦 and 𝑦 for π‘₯ in 3π‘₯βˆ’2=𝑦. This gives 3π‘¦βˆ’2=π‘₯.
  2. We rearrange the previous equation in the form 𝑦=𝑓(π‘₯).
    1. We add 2 to each side of the equation: 3𝑦=π‘₯+2.
    2. We divide each side by 3: 𝑦=π‘₯+23.
  3. We find the range of 𝑓: all the real numbers.
  4. We write our solution: 𝑓(π‘₯)=π‘₯+23.

Example 4: Finding the Inverse of a Function Algebraically

Find 𝑓(π‘₯) for 𝑓(π‘₯)=3+√π‘₯.

Answer

  1. We swap π‘₯ for 𝑦 and 𝑦 for π‘₯ in 3+√π‘₯=π‘¦οŽ’. This gives 3+βˆšπ‘¦=π‘₯.
  2. We rearrange the previous equation in the form 𝑦=𝑓(π‘₯).
    1. We take away 3 from each side of the equation: οŽ’βˆšπ‘¦=π‘₯βˆ’3.
    2. We raise each side to the power of 3: 𝑦=(π‘₯βˆ’3).
  3. We find the range of 𝑓: all the real numbers.
  4. We write our solution: 𝑓(π‘₯)=(π‘₯βˆ’3).

Example 5: Finding the Inverse of a Function Algebraically

Find 𝑓(π‘₯) for 𝑓(π‘₯)=√π‘₯+3.

Answer

  1. We swap π‘₯ for 𝑦 and 𝑦 for π‘₯ in √π‘₯+3=𝑦. This gives βˆšπ‘¦+3=π‘₯.
  2. We rearrange the previous equation in the form 𝑦=𝑓(π‘₯).
    1. We take away 3 from each side of the equation: βˆšπ‘¦=π‘₯βˆ’3.
    2. We square each side: 𝑦=(π‘₯βˆ’3).
  3. We find the range of 𝑓: the square root function gives only positive numbers, so the range of 𝑓(π‘₯)=√π‘₯+3 is all numbers greater than or equal to 3, which is written as 𝑦β‰₯3. Therefore, the domain of π‘“οŠ±οŠ§ is π‘₯β‰₯3.
  4. We write our solution: 𝑓(π‘₯)=(π‘₯βˆ’3) for π‘₯β‰₯3.

Further Comments

The diagram below shows the graph of 𝑓(π‘₯)=√π‘₯+3 and its inverse 𝑓(π‘₯)=(π‘₯βˆ’3). The function β„Ž(π‘₯)=(π‘₯βˆ’3) for all the real numbers is actually a parabola. Here, by restricting the domain to π‘₯β‰₯3, we have only half of the parabola that is the reflection of 𝑓(π‘₯) in the line 𝑦=π‘₯.

Example 6: Finding the Inverse of a Function Algebraically

Find 𝑓(π‘₯) for 𝑓(π‘₯)=(π‘₯βˆ’2)βˆ’3, where π‘₯β‰₯2.

Answer

  1. We swap π‘₯ for 𝑦 and 𝑦 for π‘₯ in (π‘₯βˆ’2)βˆ’3=π‘¦οŠ¨. This gives (π‘¦βˆ’2)βˆ’3=π‘₯.
  2. We rearrange the previous equation in the form 𝑦=𝑓(π‘₯).
    1. We add 3 to each side of the equation: (π‘¦βˆ’2)=π‘₯+3.
    2. We take the square roots of each side: π‘¦βˆ’2=√π‘₯+3.
    3. We add 2 to each side: 𝑦=√π‘₯+3+2.
  3. We find the range of 𝑓: 𝑓 is a quadratic function and it is given here in its vertex form, so we know that the parabola’s vertex is the point of coordinates (2,βˆ’3) and its minimum is βˆ’3. As the domain of 𝑓 is π‘₯β‰₯2, the function 𝑓 is actually half a parabola, as shown in the diagram below. Its range is 𝑦β‰₯βˆ’3; therefore, the domain of π‘“οŠ±οŠ§ is π‘₯β‰₯βˆ’3.
  4. We write our solution: 𝑓(π‘₯)=√π‘₯+3+2 for π‘₯β‰₯βˆ’3.

Further Comments

Note that it is not possible to find the inverse of a quadratic function if its domain is not restricted to β€œhalf” or less than β€œhalf” of the parabola. (Here, with β€œhalf” of a parabola, we mean the part of a parabola on either side of its symmetry line π‘₯=β„Ž, where β„Ž is the π‘₯-coordinate of its vertex.) Indeed, if a parabola is reflected in the line 𝑦=π‘₯, the obtained graph is not the graph of a function since one element of the domain does not correspond to only one element of the range.

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