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Use Inverse Functions

In this worksheet, students will work with inverse functions.

'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 another function. If f(x) is a function, we write its inverse as f-1(x).

 

There are three steps to 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:

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.

What is the inverse of f(x)= 2x - 1.

2y - 1

2x + 1

(x + 1)/2

(x - 1)/2

f(x) = -7x

g(x) = x/(-7)

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

Yes

No

g(x) = √(x-5)

Find g-1(4).

21

9

13

-21

Match functions to the inverses.

Column A

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

f(x) = 2x + 3

Find f-1(2x+1).

2y - 3

2x - 3

(x - 3)/2

x - 1

f(x) = 2x² - 5

What is f-1(x)?

2x² + 5

√(x + 5)/2

√(x - 5)/2

(√x + 5)/2

g(x) = 3x - 4

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

x=2.6

x=17

x=-1

x=0

f(x) = x3 + 1.

Find f-1(28).

f(x) = (4x - 3)/2

Match the values of the inverse functions to the correct expressions.

Column A

Column B

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

Find the inverse of f(x) = (2 + 3x)/(x - 2).

(2 - 3x)/(x + 2)

(2 + 2x)/(x - 3)

(2 - 2x)/(x + 3)

(x - 2)/(2 + 3x)

  • Question 1

What is the inverse of f(x)= 2x - 1.

CORRECT ANSWER
(x + 1)/2
EDDIE SAYS
Follow the three steps for finding the inverse function. Step 1. y = 2x - 1 Step 2. x = 2y - 1 Step 3. x + 1 = 2y (x + 1)/2 = y f-1(x)=(x + 1)/2
  • Question 2

f(x) = -7x

g(x) = x/(-7)

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

CORRECT ANSWER
Yes
EDDIE SAYS
f-1 = x/(-7) so g(x) is an inverse of f(x).
  • Question 3

g(x) = √(x-5)

Find g-1(4).

CORRECT ANSWER
21
EDDIE SAYS
First find g-1(x). y = √(x-5) x = √(y-5) x² + 5 = y g-1(x) = x² + 5 Now substitute 4 for the x. (4)² + 5 = 16 + 5 = 21
  • Question 4

Match functions to the inverses.

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
Follow the steps discussed in the introduction as these will help you to find the inverse functions. 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 Step 1. y = 49x² Step 2. x = 49y² Step 3. y = √x/7 So f-1= √x/7.
  • Question 5

f(x) = 2x + 3

Find f-1(2x+1).

CORRECT ANSWER
x - 1
EDDIE SAYS
Take the steps to find the inverse first. y = 2x + 3 x = 2y + 3 x - 3 =2y (x - 3)/2 = y f-1=(x - 3)/2 Now replace x with 2x + 1. (2x + 1 - 3)/2 = (2x - 2)/2 = x - 1
  • Question 6

f(x) = 2x² - 5

What is f-1(x)?

CORRECT ANSWER
√(x + 5)/2
EDDIE SAYS
The inverse of f(x) = 2x² - 5 is √(x + 5)/2. y = 2x² - 5 x = 2y² - 5 x + 5 = 2y² (x + 5)/2 = y² √(x + 5)/2 = y f-1(x)=√(x + 5)/2
  • Question 7

g(x) = 3x - 4

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

CORRECT ANSWER
x=17
EDDIE SAYS
You need to find the inverse of g(x) first. y = 3x - 4 x = 3y - 4 x + 4 = 3y (x + 4)/3 = y So g-1 = (x + 4)/3. You want this to be equal to 7, so 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

f(x) = x3 + 1.

Find f-1(28).

CORRECT ANSWER
3
EDDIE SAYS
Find f-1(x) first. y = x³ + 1 x = y³ + 1 x - 1 = y³ ∛(x - 1) = y f-1(x) = ∛(x - 1). Now substitute 28 for x. ∛(28 - 1) = ∛27 = 3
  • Question 9

f(x) = (4x - 3)/2

Match the values of the inverse functions to the correct expressions.

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. Substitute the appropriate values into this inverse function and simplify. Have a look at this one: f-1(x-1) = (2(x-1)+3)/4 = (2x - 2 + 3)/4 = (2x + 1)/4
  • Question 10

Find the inverse of f(x) = (2 + 3x)/(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 on 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)
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