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Use Inverse Functions

In this worksheet, students find the inverse of functions using the method of finding an alternative subject.

'Use Inverse Functions' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Language of Functions

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

An inverse function reverses a previous function.

So if f(x) is a function, we write its inverse as f-1(x).

 

There are three steps to follow when finding an inverse function:

Step 1: Change f(x) = into y =.

Step 2: Swap x and y.

Step 3: Make y the subject.

 

 

Let's have a look at an example now.

 

e.g. Find the inverse of f(x) = 3x + 4.

 

Step 1: y = 3x + 4

Step 2: x = 3y + 4

Step 3: x - 4 = 3y

(x - 4) ÷ 3 = y

 

So f-1(x) = (x - 4) ÷ 3

 

 

 

In this activity, we will find the inverse of functions using the method shown above of finding an alternative subject of the function. 

Find the inverse of:

 

f(x)= 2x - 1

2y - 1

2x + 1

(x + 1) ÷ 2

(x - 1) ÷ 2

Consider the functions below:

 

f(x) = -7x

g(x) = x ÷ (-7)

 

Is g(x) the inverse of f(x)?

Yes

No

Consider the function below:

 

g(x) = √(x-5)

 

Find g-1(4).

21

9

13

-21

Match each function below to its inverse.

Column A

Column B

f(x) = 3x - 7
f-1=2(x - 3)
f(x) = 49x²
f-1= √x/7
f(x) = 1/2x + 3
f-1= (x + 7)/3
f(x) = x/6 + 4
f-1= 6(x - 4)

Consider the function below:

 

f(x) = 2x + 3

 

Find f-1(2x + 1).

2y - 3

2x - 3

(x - 3) ÷ 2

x - 1

Consider the function below:

 

f(x) = 2x² - 5

 

Find f-1(x).

2x2 + 5

√(x + 5) ÷ 2

√(x - 5) ÷ 2

(√x + 5) ÷ 2

Consider the function below: 

 

g(x) = 3x - 4

 

Find the value of x for which g-1(x) = 7.

x = 2.6

x = 17

x = -1

x = 0

Consider the function below:

 

f(x) = x<sup>3</sup> + 1

 

Find f-1(28).

Consider the function below:

 

f(x) = (4x - 3) &divide; 2

 

Match each values of the inverse function below to its correct expression.

Column A

Column B

f-1(2x)
(8x + 3) ÷ 2
f-1(x²)
(4x² + 3) ÷ 2
f-1(x - 1)
(2x + 1) ÷ 4
f-1(x + 1)
(2x + 5) ÷ 4

Find the inverse of:

 

f(x) = (2 + 3x) &divide; (x - 2)

(2 - 3x) ÷ (x + 2)

(2 + 2x) ÷ (x - 3)

(2 - 2x) ÷ (x + 3)

(x - 2) ÷ (2 + 3x)

  • Question 1

Find the inverse of:

 

f(x)= 2x - 1

CORRECT ANSWER
(x + 1) ÷ 2
EDDIE SAYS
Follow our three steps for finding the inverse function: Step 1: Change f(x) = into y =. y = 2x - 1 Step 2: Swap x and y. x = 2y - 1 Step 3: Make y the subject. x + 1 = 2y (x + 1) ÷ 2 = y So f-1(x)=(x + 1) ÷ 2 How did you get on with that? Review the Introduction before you move on if you found that tricky at all.
  • Question 2

Consider the functions below:

 

f(x) = -7x

g(x) = x ÷ (-7)

 

Is g(x) the inverse of f(x)?

CORRECT ANSWER
Yes
EDDIE SAYS
To find out if this statement is true or not, we need to work out the inverse of f(x). Let's follow our three-step process again. 1) Change f(x) = into y =. y = -7x 2) Swap x and y. -7x = y Step 3: Make y the subject. x = y ÷ -7 So f-1 = x ÷ (-7) This is the same as g(x), which means this is an inverse of f(x). Did you spot that?
  • Question 3

Consider the function below:

 

g(x) = √(x-5)

 

Find g-1(4).

CORRECT ANSWER
21
EDDIE SAYS
This question is asking us to find the inverse of g(x) then to substitute 4 for g in the function. First let's find g-1(x). y = √(x-5) x = √(y-5) x² + 5 = y So g-1(x) = x² + 5 Now let's substitute 4 for the x. (4)² + 5 = 16 + 5 = 21
  • Question 4

Match each function below to its inverse.

CORRECT ANSWER

Column A

Column B

f(x) = 3x - 7
f-1= (x + 7)/3
f(x) = 49x²
f-1= √x/7
f(x) = 1/2x + 3
f-1=2(x - 3)
f(x) = x/6 + 4
f-1= 6(x - 4)
EDDIE SAYS
We need to follow the steps discussed in the Introduction to help us find the inverse functions of each function. Step 1: Write f(x) = as y = Step 2: Swap x and y Step 3: Make y the subject Let's have a look at f(x) = 49x² as an example. Step 1: y = 49x² Step 2: x = 49y² Step 3: y = √x/7 So f-1= √x/7 Can you find the inverse functions and match the other three pairs independently, using this example to help?
  • Question 5

Consider the function below:

 

f(x) = 2x + 3

 

Find f-1(2x + 1).

CORRECT ANSWER
x - 1
EDDIE SAYS
Here we need to find the inverse then substitute '2x + 1' for x in the function. Let's follow the steps to find the inverse first: y = 2x + 3 x = 2y + 3 x - 3 =2y (x - 3) ÷ 2 = y So f-1 = (x - 3) ÷ 2 Now let's replace 'x' with '2x + 1': (2x + 1 - 3) ÷ 2 = (2x - 2) ÷ 2 = x - 1
  • Question 6

Consider the function below:

 

f(x) = 2x² - 5

 

Find f-1(x).

CORRECT ANSWER
√(x + 5) ÷ 2
EDDIE SAYS
Let's follow our steps to find the inverse function here. 1) y = 2x² - 5 2) x = 2y² - 5 3) x + 5 = 2y² (x + 5) ÷ 2 = y² √(x + 5) ÷ 2 = y So f-1(x)=√(x + 5) ÷ 2 Did you remember that the inverse of squaring a number is to find the square root?
  • Question 7

Consider the function below: 

 

g(x) = 3x - 4

 

Find the value of x for which g-1(x) = 7.

CORRECT ANSWER
x = 17
EDDIE SAYS
To start with, we need to find the inverse of g(x). y = 3x - 4 x = 3y - 4 x + 4 = 3y (x + 4) ÷ 3 = y So g-1 = (x + 4) ÷ 3 Now we need to find the value of x which means this function equals 7. To do this we need to solve the equation: (x + 4) ÷ 3 = 7 x + 4 = 3 × 7 x + 4 = 21 x = 21 - 4 x = 17 So the value of x for which g-1(x) = 7 is 12.
  • Question 8

Consider the function below:

 

f(x) = x<sup>3</sup> + 1

 

Find f-1(28).

CORRECT ANSWER
3
EDDIE SAYS
Let's ind f-1(x) first. y = x³ + 1 x = y³ + 1 x - 1 = y³ ∛(x - 1) = y f-1(x) = ∛(x - 1) Now let's substitute 28 for x: ∛(28 - 1) = ∛27 = 3
  • Question 9

Consider the function below:

 

f(x) = (4x - 3) &divide; 2

 

Match each values of the inverse function below to its correct expression.

CORRECT ANSWER

Column A

Column B

f-1(2x)
(8x + 3) ÷ 2
f-1(x²)
(4x² + 3) ÷ 2
f-1(x - 1)
(2x + 1) ÷ 4
f-1(x + 1)
(2x + 5) ÷ 4
EDDIE SAYS
The inverse of f(x) here is: (2x + 3) ÷ 4 Then we need to substitute the appropriate values into this inverse function and simplify our outcomes. Let's take a look at one example together: f-1(x - 1) = (2(x - 1) + 3) ÷ 4 = (2x - 2 + 3) ÷ 4 = (2x + 1) ÷ 4 Can you substitute the other values into the inverse function to find their matching expressions?
  • Question 10

Find the inverse of:

 

f(x) = (2 + 3x) &divide; (x - 2)

CORRECT ANSWER
(2 + 2x) ÷ (x - 3)
EDDIE SAYS
There's a bit of algebraic manipulation to do here! y = (2 + 3x) ÷ (x - 2) x = (2 + 3y) ÷ (y - 2) Now multiply by (y - 2) to remove the fraction: x(y - 2) = (2 + 3y) xy - 2x = (2 + 3y) Move all the terms with y to one side: xy - 3y = 2 + 2x Factorise y out and then make it the subject: y(x - 3) = 2 + 2x y = (2 + 2x) ÷ (x - 3) So f-1 = (2 + 2x) ÷ (x - 3) Phew! That was a tricky one to finish with; well done if you got it right!
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