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Composite Functions
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Q1. S21/qp12/q5
The function 𝑓 is defined by 𝑓(𝑥) = 2𝑥2 + 3 for 𝑥 ≥ 0.
a. Find and simplify an expression for 𝑓𝑓(𝑥). [2]
b. Solve the equation 𝑓𝑓(𝑥) = 34𝑥 2 + 19. [4]
Q2. S21/qp13/q8(a)
Functions 𝑓 and 𝑔 are defined as follows:
𝑓 ∶ 𝑥 → 𝑥2 − 1 for 𝑥 < 0,
1 1
𝑔∶ 𝑥 → for 𝑥 < −2 .
2𝑥+1
Solve the equation 𝑓𝑔(𝑥) = 3. [4]
Q3. W20/qp12/q5(a)
4
Functions 𝑓 and 𝑔 are defined by 𝑓(𝑥) = 4𝑥 − 2, for 𝑥 ∈ ℝ, 𝑔(𝑥) = , for x ∈ ℝ , 𝑥 ≠ −1.
𝑥+1
Find the value of 𝑓𝑔(7). [1]
Q4. W19/qp11/q7(i,ii)
Functions 𝑓 and 𝑔 are defined by
3
𝑓 ∶ 𝑥 → 2𝑥+1 for 𝑥 > 0,
1
𝑔 ∶ 𝑥 → 𝑥 + 2 for 𝑥 > 0.
i. Find the range of 𝑓 and the range of 𝑔. [3]
𝑎𝑥
ii. Find an expression for 𝑓𝑔(𝑥), giving your answer in the form 𝑏𝑥+𝑐, where 𝑎, 𝑏 and 𝑐 are
integers. [2]
Q5. S19/qp12/q7(ii)
2𝑥 + 3
Functions 𝑓 and 𝑔 are defined by 𝑓 ∶ 𝑥 → 3𝑥 − 2, 𝑥 ∈ ℝ >, 𝑔 ∶ 𝑥 → , 𝑥 ∈ ℝ, 𝑥 ≠ 1
𝑥−1
7
Solve the equation 𝑓𝑔(𝑥) = 3. [3]
Q6. S19/qp13/q4(ii)
48
The function 𝑓 is defined by 𝑓(𝑥) = for 3 ≤ 𝑥 ≤ 7. The function 𝑔 is defined by 𝑔(𝑥) = 2𝑥 − 4
𝑥−1
Find an expression for 𝑔𝑓(𝑥). [1]
Q7. W18/qp12/q4
Functions f and g are defined by 𝑓 ∶ 𝑥 → 2 − 3 𝑐𝑜𝑠 𝑥 for 0 ≤ 𝑥 ≤ 20,
1
𝑔∶ 𝑥 → 𝑥 for 0 ≤ 𝑥 ≤ 2𝜋.
2
i. Solve the equation 𝑓𝑔(𝑥) = 1. [3]
ii. Sketch the graph of 𝑦 = 𝑓(𝑥). [3]
Q8. W18/qp11/q11(b)
The function 𝑔 is defined by 𝑔(𝑥) = (𝑥 − 3)2 for 𝑥 ≥ 0.
i. Show that 𝑔𝑔(2𝑥) can be expressed in the form (2𝑥 – 3)4 + 𝑏(2𝑥 – 3)2 + 𝑐, where 𝑏 and
𝑐 are constants to be found. [2]
ii. Hence expand 𝑔𝑔(2𝑥) completely, simplifying your answer. [4]
Q9. S18/qp11/q9(iii)
1 1
Functions f and g are defined for x ∈ ℝ by 𝑓 ∶ 𝑥 → 𝑥 − 2, 𝑔 ∶ 𝑥 → 4 + 𝑥 − 𝑥2 .
2 2
Find an expression for 𝑓𝑔(𝑥) and deduce the range of 𝑓𝑔. [4]
Q10. W14/qp13/q10(a)(i)
1
The functions 𝑓 and 𝑔 are defined for 𝑥 ≥ 0 by 𝑓 ∶ 𝑥 → (𝑎𝑥 + 𝑏)3 , where 𝑎 and 𝑏 are positive
constants, 𝑔 ∶ 𝑥 → 𝑥2 . Given that 𝑓𝑔(1) = 2 and 𝑔𝑓(9) = 16,
Calculate the values of 𝑎 and 𝑏. [4]
Answers
(Composite Functions)
Q1. S21/qp12/q5
Q2. S21/qp13/q8(a)
Q3. W20/qp12/q5(a)
Q4. W19/qp11/q7(I,ii)
Q5. S19/qp12/q7(ii)
Q6. S19/qp13/q4(ii)
Q7. W18/qp12/q4
Q8. W18/qp11/q11(b)
Q9. S18/qp11/q9(iii)
Q10. W14/qp13/q10(a)(i)
