Functions (Copy)
Question 1
Let f(x) = 3x – 5.
(a) Find f(4).
(b) State the domain and range if x ∈ {1, 2, 3, 4, 5}.
Question 2
f(x) = 2x + 7.
Find f(–3) and f(0).
Question 3
Let f(x) = 5x – 2.
Find the inverse function f⁻¹(x).
Question 4
Let f(x) = (x – 4)/3.
Find f⁻¹(x).
Question 5
f(x) = 2x + 1 and g(x) = x².
Find:
(a) fg(x)
(b) gf(x).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
f(x) = x² + 2 and g(x) = 3x.
Find fg(2) and gf(2).
Question 7
f(x) = 3x – 1.
Draw the mapping diagram if x ∈ {1,2,3,4}.
Question 8
f(x) = (x + 2)².
Find f(–3), f(0), f(2).
Question 9
If f(x) = 4x + 3 and g(x) = x – 1,
find fg(x) and gf(x).
Question 10
f(x) = (2x – 1)/3.
Find f⁻¹(x).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
f(x) = 3x – 5.
Find the domain and range if x is any integer.
Question 12
If f(x) = 1/x, state the domain and range.
Question 13
Let f(x) = 2x and g(x) = x + 5.
Find gf(3) and fg(3).
Question 14
f(x) = (x – 1)².
Find f⁻¹(x) (principal square root only).
Question 15
f(x) = 3x + 1, g(x) = 5x.
Find fg(x) and gf(x).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
f(x) = 7 – 2x.
Find f⁻¹(x).
Question 17
f(x) = 2x + 1.
Find f(f(3)).
Question 18
f(x) = 2x – 5, g(x) = (x + 3)/2.
Find gf(x).
Question 19
f(x) = √(x + 4).
State the domain and range.
Question 20
f(x) = x³.
Find f⁻¹(x).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 21
f(x) = (x – 2)/(x + 1).
Find f(1) and state restrictions on the domain.
Question 22
f(x) = 3x² – 1.
Find the range if domain is {–2, –1, 0, 1, 2}.
Question 23
f(x) = 4x – 7, g(x) = 2x + 5.
Find gf(x) and fg(x).
Question 24
If f(x) = (x + 1)/2, show that f⁻¹(x) = 2x – 1.
Question 25
f(x) = 5x – 4, g(x) = x².
Find fg(x).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 26
f(x) = 2x, g(x) = x – 3.
Find gf(x).
Question 27
f(x) = x + 4, g(x) = 1/x.
Find fg(x) and gf(x).
Question 28
f(x) = (x – 1)², g(x) = √x.
Find gf(x).
Question 29
f(x) = 2x + 3, g(x) = 4 – x.
Find fg(2) and gf(2).
Question 30
f(x) = 3x – 2.
Find f(f(f(2))).
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
1.
f(x) = 3x – 5
(a) f(4) = 3(4) – 5 = 12 – 5 = 7.
(b) Domain = {1,2,3,4,5}. Range = {–2,1,4,7,10}.
2.
f(x) = 2x + 7
f(–3) = 2(–3) + 7 = –6 + 7 = 1.
f(0) = 2(0) + 7 = 7.
3.
f(x) = 5x – 2
Let y = 5x – 2.
x = (y + 2)/5.
So f⁻¹(x) = (x + 2)/5.
4.
f(x) = (x – 4)/3
y = (x – 4)/3 → x = 3y + 4.
f⁻¹(x) = 3x + 4.
5.
f(x) = 2x + 1, g(x) = x²
fg(x) = f(x²) = 2(x²) + 1 = 2x² + 1.
gf(x) = g(2x + 1) = (2x + 1)² = 4x² + 4x + 1.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
f(x) = x² + 2, g(x) = 3x
fg(2) = f(g(2)) = f(6) = 6² + 2 = 38.
gf(2) = g(f(2)) = g(6) = 3(6) = 18.
7.
f(x) = 3x – 1
Mapping: {1→2, 2→5, 3→8, 4→11}.
8.
f(x) = (x + 2)²
f(–3) = (–1)² = 1.
f(0) = (2)² = 4.
f(2) = (4)² = 16.
9.
f(x) = 4x + 3, g(x) = x – 1
fg(x) = f(g(x)) = f(x – 1) = 4(x – 1) + 3 = 4x – 1.
gf(x) = g(f(x)) = g(4x + 3) = (4x + 3) – 1 = 4x + 2.
10.
f(x) = (2x – 1)/3
y = (2x – 1)/3 → 3y = 2x – 1 → x = (3y + 1)/2.
f⁻¹(x) = (3x + 1)/2.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11.
f(x) = 3x – 5, x ∈ Z
Domain = all integers.
Range = all integers (since linear with integer coefficients).
12.
f(x) = 1/x
Domain: all real x except 0.
Range: all real y except 0.
13.
f(x) = 2x, g(x) = x + 5
gf(3) = g(f(3)) = g(6) = 11.
fg(3) = f(g(3)) = f(8) = 16.
14.
f(x) = (x – 1)²
Inverse: y = (x – 1)² → √y = x – 1 → x = √y + 1.
So f⁻¹(x) = √x + 1 (principal root).
15.
f(x) = 3x + 1, g(x) = 5x
fg(x) = f(g(x)) = f(5x) = 3(5x) + 1 = 15x + 1.
gf(x) = g(f(x)) = g(3x + 1) = 5(3x + 1) = 15x + 5.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
f(x) = 7 – 2x
y = 7 – 2x → 2x = 7 – y → x = (7 – y)/2.
f⁻¹(x) = (7 – x)/2.
17.
f(x) = 2x + 1
f(f(3)) = f(7) = 15.
18.
f(x) = 2x – 5, g(x) = (x + 3)/2
gf(x) = g(2x – 5) = (2x – 5 + 3)/2 = (2x – 2)/2 = x – 1.
19.
f(x) = √(x + 4)
Domain: x ≥ –4.
Range: y ≥ 0.
20.
f(x) = x³
Inverse: y = x³ → x = ³√y.
f⁻¹(x) = ³√x.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
21.
f(x) = (x – 2)/(x + 1)
f(1) = (1 – 2)/(1 + 1) = –1/2.
Domain restriction: x ≠ –1.
22.
f(x) = 3x² – 1
Domain {–2, –1, 0, 1, 2}.
Range = {11,2,–1,2,11}.
23.
f(x) = 4x – 7, g(x) = 2x + 5
gf(x) = g(4x – 7) = 2(4x – 7) + 5 = 8x – 9.
fg(x) = f(2x + 5) = 4(2x + 5) – 7 = 8x + 13.
24.
f(x) = (x + 1)/2
y = (x + 1)/2 → x = 2y – 1.
So f⁻¹(x) = 2x – 1.
25.
f(x) = 5x – 4, g(x) = x²
fg(x) = f(x²) = 5x² – 4.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
26.
f(x) = 2x, g(x) = x – 3
gf(x) = g(2x) = 2x – 3.
27.
f(x) = x + 4, g(x) = 1/x
fg(x) = f(1/x) = 1/x + 4.
gf(x) = g(x + 4) = 1/(x + 4).
28.
f(x) = (x – 1)², g(x) = √x
gf(x) = g((x – 1)²) = √((x – 1)²) = |x – 1|.
29.
f(x) = 2x + 3, g(x) = 4 – x
fg(2) = f(g(2)) = f(2) = 7.
gf(2) = g(f(2)) = g(7) = –3.
30.
f(x) = 3x – 2
f(2) = 4.
f(f(2)) = f(4) = 10.
f(f(f(2))) = f(10) = 28.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course