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7.9D Inverse and Composite Functions: Fluency and Exam Drill

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 50 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Work out f(2) for f(x) = 3x + 1.

(Total for Question 1 is 1 mark)

Work out g(3) for g(x) = x² - 1.

(Total for Question 2 is 1 mark)

Work out h(0) for h(x) = 4x - 5.

(Total for Question 3 is 1 mark)

Work out f(-2) for f(x) = 2x².

(Total for Question 4 is 1 mark)

Let f(x) = x + 4 and g(x) = 3x.
Find (f o g)(2).

(Total for Question 5 is 2 marks)

Let f(x) = x² and g(x) = x + 1.
Find (g o f)(3).

(Total for Question 6 is 2 marks)

Given f(x) = 2x + 1 and g(x) = x², write down (f o g)(x) in simplified form.

(Total for Question 7 is 2 marks)

Let f(x) = x - 5 and g(x) = 3x + 2.
(a) Find (g o f)(x).
(b) Find (f o g)(x).

(a)Find (g o f)(x).(2)

(b)Find (f o g)(x).(2)

(Total for Question 8 is 4 marks)

Find the inverse function f^{-1}(x) for f(x) = 3x + 4. Give the inverse in terms of x.

(Total for Question 9 is 2 marks)

Find the inverse function f^{-1}(x) for f(x) = (x + 6)/4.

(Total for Question 10 is 2 marks)

Show that the inverse of f(x) = 5x - 8 is f^{-1}(x) = (x + 8)/5. Give full working.

(Total for Question 11 is 3 marks)

Let f(x) = 2x + 3.
(a) Find f^{-1}(x).
(b) Verify that (f o f^{-1})(x) = x.

(a)Find f^{-1}(x).(2)

(b)Verify that (f o f^{-1})(x) = x.(1)

(Total for Question 12 is 3 marks)

Find the inverse function f^{-1}(x) for f(x) = (x - 1)/(x + 2). Give the answer in simplified form and state any restriction on x for f^{-1}.

(Total for Question 13 is 4 marks)

Let f(x) = 3x - 4.
(a) Find f^{-1}(x).
(b) Calculate (f^{-1} o f)(5).

(a)Find f^{-1}(x).(2)

(b)Calculate (f^{-1} o f)(5).(1)

(Total for Question 14 is 3 marks)

Let f(x) = 4x - 7 and g(x) = x².
Solve (f o g)(x) = 17. Give all real solutions.

(Total for Question 15 is 2 marks)

Let f(x) = (2x + 1)/3 and g(x) = 3x - 1.
Find (f^{-1} o g)(2).

(Total for Question 16 is 3 marks)

Mark scheme · 7.9D Inverse and Composite Functions: Fluency and Exam Drill

Question 1

B1 7 cao
Answer: 7

Question 2

B1 8 cao
Answer: 8

Question 3

B1 -5 cao
Answer: -5

Question 4

B1 8 cao
Answer: 8

Question 5

M1 substitute g(2)=3*2=6 and use f(6)=6+4 or equivalent method
A1 10 cao
Answer: 10

Question 6

M1 compute f(3)=9 then g(9)=9+1 or equivalent
A1 10 cao
Answer: 10

Question 7

M1 substitute g(x)=x² into f to get 2*(x²)+1 or equivalent
A1 2x² + 1 cao
Answer: 2x² + 1

Question 8

(a) M1 substitute f(x) into g: g(f(x))=3(x-5)+2 or equivalent
(a) A1 3x - 13 cao
(a) Answer: 3x - 13
(b) M1 substitute g(x) into f: f(g(x))=(3x+2)-5 or equivalent
(b) A1 3x - 3 cao
(b) Answer: 3x - 3

Question 9

M1 method: set y=3x+4, swap x and y and solve for y or equivalent
A1 f^{-1}(x) = (x - 4)/3 cao
Answer: f^{-1}(x) = (x - 4)/3

Question 10

M1 set y=(x+6)/4, swap x and y and solve for y or equivalent
A1 f^{-1}(x) = 4x - 6 cao
Answer: f^{-1}(x) = 4x - 6

Question 11

M1 set y = 5x - 8 and swap x and y or equivalent
M1 rearrange to get x = 5y - 8 then solve y = (x + 8)/5
A1 f^{-1}(x) = (x + 8)/5 cso
Answer: f^{-1}(x) = (x + 8)/5

Question 12

(a) M1 set y = 2x + 3, swap and solve or equivalent
(a) A1 f^{-1}(x) = (x - 3)/2 cao
(a) Answer: f^{-1}(x) = (x - 3)/2
(b) B1 f(f^{-1}(x)) = x cao
(b) Answer: f(f^{-1}(x)) = x

Question 13

M1 set y = (x - 1)/(x + 2) and multiply both sides by (x + 2) or equivalent
M1 rearrange to get terms with x and y on correct sides, e.g. yx + 2y = x - 1
M1 isolate x: x(y - 1) = -(1 + 2y) and solve for x
A1 f^{-1}(x) = (1 + 2x)/(1 - x) cao; x ≠ 1 stated
Answer: f^{-1}(x) = (1 + 2x)/(1 - x); restriction x ≠ 1

Question 14

(a) M1 set y = 3x - 4, swap and solve or equivalent
(a) A1 f^{-1}(x) = (x + 4)/3 cao
(a) Answer: f^{-1}(x) = (x + 4)/3
(b) A1 5 cao
(b) Answer: 5

Question 15

M1 form equation 4x² - 7 = 17 and solve to x² = 6 or equivalent
A1 x = ±√6 cao
Answer: x = ±√6

Question 16

M1 find inverse: f^{-1}(x) = (3x - 1)/2 or equivalent
M1 compute g(2) = 3*2 - 1 = 5
A1 (f^{-1} o g)(2) = 7 cao
Answer: 7

Inverse and Composite Functions: Fluency and Exam Drill - Worksheets, Questions and Revision