C3 FUNCTIONS Answers - Worksheet C

1 a y = 10x + 3 b y = 9 + 2x c y = 5 − 6x
swap x = 10y + 3 swap x = 9 + 2y swap x = 5 − 6y
x−3 x−9 5− x
y= y= y=
10 2 6
x−3 x−9 5− x
f −1(x) = , x∈ f −1(x) = , x∈ f −1(x) = , x∈
10 2 6

x+3
d y= e y= 1
3
(2x − 5) f y=8− 3
5
x
4
y+3
swap x = swap x = 1
3
(2y − 5) swap x = 8 − 3
5
y
4
3x + 5 40 − 5 x
y = 4x − 3 y= y=
2 3
3x + 5 40 − 5 x
f −1(x) = 4x − 3, x ∈ f −1(x) = , x∈ f −1(x) = , x∈
2 3

1
2 a y = ln x b y=
x
1
swap x = ln y swap x =
y
1
y = ex y=
x
1
f −1(x) = ex, x ∈ f −1(x) = , x∈ , x≠0
x

4
c y= x d y = 3x − 4
swap x = 4 y swap x = 3y − 4
x+4
y = x4 y=
3
f −1(x) = x4, x ∈ , x > 0 f(0) = −4, f(3) = 5
x+4
f −1(x) = , x ∈ , −4 ≤ x < 5
3

1 1
e y= f y=2+
x−5 x
1 1
swap x = swap x = 2 +
y −5 y
1 1
y= +5 y=
x x−2
1 1
f −1(x) = + 5, x ∈ , x ≠ 0 f −1(x) = , x∈ , x≠2
x x−2

 Solomon Press
C3 FUNCTIONS Answers - Worksheet C page 2

1− x 10
3 a i y = 2x + 1 b i y= c i y=
5 x
1− y 10
swap x = 2y + 1 swap x = swap x =
5 y
x −1 10
y= y = 1 − 5x y=
2 x
x −1 10
f −1 : x → , x∈ f −1 : x → 1 − 5x, x ∈ f −1 : x → , x∈ ,x≠0
2 x
ii y ii y ii y
y = f(x) y = f(x) and
y = f −1(x) y = f −1(x)
y = f −1(x)
y = f(x) O x
O x O x

d i y = x2 e i y = ex f i y = x3
swap x = y2 swap x = ey swap x = y3
y=± x y = ln x y= 3 x
(domain of f ⇒ +) f −1 : x → ln x, x ∈ , x > 0 f −1 : x → 3
x, x∈
−1
f :x→ x , x ∈ , x > 0 ii y y = f(x) ii y y = f(x)
ii y y = f(x)
y = f −1(x)
−1
y = f (x)
O x O x
y = f −1(x)
O x

2x − 4
4 a y = 5x + 1 b y= c y = ex + 2
3
2y − 4
swap x = 5y + 1 swap x = swap x = ey + 2
3
x −1 3x + 4
f −1(x) = y = f −1(x) = y = f −1(x) = y = ln (x − 2)
5 2
x −1 3x + 4
=2 =7−x ln (x − 2) = ln (3x − 8)
5 2
x − 1 = 10 3x + 4 = 14 − 2x x − 2 = 3x − 8
x = 11 x=2 x=3
4
d y= x+2 e y=
x+3
4
swap x = y+2 swap x =
y+3
4
f −1(x) = y = x2 − 2 f −1(x) = y = −3
x
4
x2 − 2 = 3x − 4 − 3 = 5(x + 1)
x
x2 − 3x + 2 = 0 4 − 3x = 5x(x + 1)
(x − 1)(x − 2) = 0 5x2 + 8x − 4 = 0
x = 1, 2 (5x − 2)(x + 2) = 0
x = −2, 25

 Solomon Press
C3 FUNCTIONS Answers - Worksheet C page 3

1
5 a y 6 a g ⇒ y=
2x + 4
1
y = f(x) swap x =
2y + 4
1 1
y = f −1(x) y= 1
2
( − 4) = −2
x 2x
1
O x g −1(x) = − 2, x ∈ , x ≠ 0
2x
range: g −1(x) ∈ , g −1(x) ≠ −2
b 4 − 2x = x b = g(3 − 2x)
4 1 1
x= 3
= =
2(3 − 2 x) + 4 10 − 4x
1
∴ ( 43 , 4
3
) gf(x) = , x∈ , x≠ 5
2
10 − 4x
c f ⇒ y = 3 − 2x
swap x = 3 − 2y
3− x
f −1(x) = y =
2
1 3− x
∴ =
10 − 4x 2
2 = (3 − x)(10 − 4x)
2x2 − 11x + 14 = 0
(2x − 7)(x − 2) = 0
x = 2, 72

1 5
7 a i y = 5x + 2 ii = f( ) iii y = +2
x x
5 5
swap x = 5y + 2 = +2 swap x = +2
x y
x−2 5 5
y= fg : x → + 2, x ∈ , x ≠ 0 y=
5 x x−2
x−2 5
f −1 : x → , x∈ (fg)−1 : x→ ,x∈ ,x≠2
5 x−2

x−2 5
b = +2
5 x
x(x − 2) = 25 + 10x
x2 − 12x − 25 = 0
12 ± 144 + 100
x= =6± 61 = −1.81, 13.81
2

 Solomon Press
C3 FUNCTIONS Answers - Worksheet C page 4

x−2
8 a y= 1
2
ln (4x − 9) b y=
x+5
y−2
swap x = 1
2
ln (4y − 9) swap x =
y+5
4y − 9 = e2x xy + 5x = y − 2
y = 14 (e2x + 9) y(1 − x) = 5x + 2
5x + 2
f −1 : x → 1
4
(e2x + 9), x ∈ y=
1− x
5x + 2
f −1 : x → , x∈ , x≠1
1− x

c y = e0.4x − 2 d y= 3
x5 − 3
swap x = e0.4y − 2 swap x = 3
y5 − 3
0.4y − 2 = ln x y5 − 3 = x3
5
y= 5
2
(2 + ln x) y= x3 + 3
f −1 : x → 5 + 5
2
ln x, x ∈ , x > 0 f −1 : x → 5
x3 + 3 , x ∈

4−x
e y = log10 (2 − 7x) f y=
3x + 2
4− y
swap x = log10 (2 − 7y) swap x =
3y + 2
2 − 7y = 10x 3xy + 2x = 4 − y
y = 17 (2 − 10x) y(3x + 1) = 4 − 2x
4 − 2x
f −1 : x → 1
7
(2 − 10x), x ∈ y=
3x + 1
4 − 2x
f −1 : x → , x ∈ , x ≠ − 13
3x + 1

9 a i y = e2x b i y = x2 + 4
swap x = e2y swap x = y2 + 4
2y = ln x y = ± x−4
y = 12 ln x (domain of f ⇒ +)
f −1 : x → 1
2
ln x, x ∈ , x > 0 f −1 : x → x−4, x ∈ , x > 4
ii y ii y
y = f(x) y = f(x)

y = f −1(x) y = f −1(x)

O x O x

 Solomon Press
C3 FUNCTIONS Answers - Worksheet C page 5

c i y = ln (x − 3) d i y = x2 + 6x + 9 = (x + 3)2
swap x = ln (y − 3) swap x = (y + 3)2
y − 3 = ex y = −3 ± x
y = ex + 3 (domain of f ⇒ +)
f −1 : x → ex + 3, x ∈ f −1 : x → −3 + x , x ∈ , x > 0
ii y y = f −1(x) ii y

y = f(x)
y = f(x)

O x O x
y = f −1(x)

10 a i f(x) = (x + 3)2 − 6 b i f(x) = (x − 2)2 + 1

x < −3 ∴ range: f(x) > −6 x ≥ 2 ∴ range: f(x) ≥ 1
ii y = (x + 3)2 − 6 ii y = (x − 2)2 + 1
swap x = (y + 3)2 − 6 swap x = (y − 2)2 + 1
y = −3 ± x + 6 y = 2 ± x −1
(domain of f ⇒ −) (domain of f ⇒ +)
f −1(x) = −3 − x + 6 , x ∈ , x > −6 f −1(x) = 2 + x − 1 , x ∈ , x ≥ 1

5 2 3 2
c i f(x) = (x + 2
) − 33
4
d i f(x) = (x − 2
) + 11
4

x < − 52 ∴ range: f(x) > −8 14 2 < x < 4, f(2) = 3, f(4) = 9

5 2
ii y = (x + 2
) − 33
4
∴ range: 3 < f(x) < 9
5 2 3 2
swap x = (y + 2
) − 33
4
ii y = (x − 2
) + 11
4
3 2
y = − 52 ± x + 334 swap x = (y − 2
) + 11
4

(domain of f ⇒ −) y= 3
2
± x − 114
f −1(x) = − 52 − x + 334 , x ∈ , x > −8 14 (domain of f ⇒ +)
f −1(x) = 3
2
+ x − 114 , x ∈ , 3 < x < 9

e i f(x) = 8 − 2x − x2 = 9 − (x + 1)2 f i f(x) = −5(x2 − 4x) = 20 − 5(x − 2)2

x ≥ −1 ∴ range: f(x) ≤ 9 x > 2 ∴ range: f(x) < 20
ii y = 9 − (x + 1)2 ii y = 20 − 5(x − 2)2
swap x = 9 − (y + 1)2 swap x = 20 − 5(y − 2)2
20 − x
y = −1 ± 9− x y=2±
5
(domain of f ⇒ +) (domain of f ⇒ +)
20 − x
f −1(x) = −1 + 9− x , x ∈ , x ≤ 9 f −1(x) = 2 + , x ∈ , x < 20
5

 Solomon Press
C3 FUNCTIONS Answers - Worksheet C page 6

x+3
11 a y= 1
3
(2x − 5) b y = ln c y = x2 − 4
5
y+3
swap x = 1
3
(2y − 5) swap x = ln swap x = y2 − 4
5
3x + 5
f −1(x) = y = f −1(x) = y = 5ex − 3 y= ± x+4
2
3x + 5 4
∴ = ∴ 5ex − 3 = 10 − 6e−x (domain of f ⇒ +)
2 2− x
(3x + 5)(2 − x) = 8 5e2x − 13ex + 6 = 0 f −1(x) = x+4
2 x x x+6
3x − x − 2 = 0 (5e − 3)(e − 2) = 0 ∴ x+4 =
3
( x + 6)2
(3x + 2)(x − 1) = 0 ex = 3
5
,2 x+4=
9
x = − 23 , 1 x = ln 3
5
, ln 2 9(x + 4) = x2 + 12x + 36
x = −0.511, 0.693 (3sf) x2 + 3x = 0
x(x + 3) = 0
x = −3, 0

12 a −2 13 a y = x2 − 3x = (x − 3 2
2
) − 9
4
6+b 3 2
b =4 swap x = (y − 2
) − 9
4
6−2
6 + b = 16 y= 3
2
± x + 94
b = 10 (domain of f ⇒ +)
x + 10
c y= f −1 : x → 3
2
+ x + 94 , x ∈ , x ≥ − 94
x−2
y + 10
swap x = b y
y−2
xy − 2x = y + 10 y = f −1(x)
y(x − 1) = 2x + 10
2 x + 10
y= y = f(x)
x −1
2 x + 10
f −1(x) = , x∈ , x≠1 O x
x −1

c g ⇒ y = 2x + 3
swap x = 2y + 3
x−3
g −1(x) = y =
2
g −1(12) = 9
2

f −1g−1(12) = f −1( 92 )
3 27
= 2
+ 4
3 3
= 2
+ 2
3
3
= 2
(1 + 3)
∴ a= 1 12

 Solomon Press