 # Use Inverse Functions

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

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

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 B

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

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

(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)?

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).

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.

## 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).

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).

√(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.

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).

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.

## 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)

(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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