Ant. / Ans. 1.1
f(x) = 2x + 5
y = 2x + 5 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 2y + 5 . . . Ruil x en y om / Swop x and y
2y = x − 5
1
5
y = ── x − ──
2
2
1
5
f
−1(x) = ── x − ──
2
2
Vr / Qu 1.1
Ant. / Ans. 1.2
f(x) = 4x − 7
y = 4x − 7 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 4y − 7 . . . Ruil x en y om / Swop x and y
4y = x + 7
1
7
y = ── x − ──
4
4
1
7
f
−1(x) = ── x − ──
4
4
Vr / Qu 1.2
Ant. / Ans. 1.3
f(x) = 3 − 2x
y = 3 − 2x . . . Vervang f(x) met y /
Substitute y for f(x)
x = 3 − 2y . . . Ruil x en y om / Swop x and y
2y = 3 − x
3
1
y = ── − ── x
2
2
3
1
f
−1(x) = ── − ── x
2
2
Vr / Qu 1.3
Ant. / Ans. 1.4
f(x) = − 5x − 6
y = − 5x − 6 . . . Vervang f(x) met y /
Substitute y for f(x)
x = − 5y − 6 . . . Ruil x en y om / Swop x and y
5y = − x − 6
1
6
y = − ── x − ──
5
5
1
6
f
−1(x) = − ── x − ──
5
5
Vr / Qu 1.4
2.1
f(x) = 3x − 9
y = 3x − 9 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 3y − 9 . . . Ruil x en y om / Swop x and y
3y = x + 9
1
y = ── x + 3
3
2.2 f(x) = 3x − 9
y = 3x − 9 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 3y − 9 . . . Ruil x en y om / Swop x and y
3y = x + 9
1
y = ── x + 3
3
grafieke / graphs
1
Y-afsnitte / Y-intercepts :
f : y = 3(0) − 9 ||
g : y = ── (0) + 3
3
(0 ; − 9) || (0 ; 3)
1
X-afsnitte / X-intercepts :
f : 0 = 3x − 9 ||
g : 0 = ── x + 3
3
x = 3 ||
x = − 9
(3 ; 0) ||
(− 9 ; 0)
Vraag / Question 2
3.1
f(x) = − 2x + 6
y = − 2x + 6 . . . Vervang f(x) met y /
Substitute y for f(x)
x = − 2y + 6 . . . Ruil x en y om / Swop x and y
2y = −x + 6
1
y = − ── x + 3
2
3.2 f(x) = −2x + 6 of / or
y = −2x + 6
1
1
f−1(x) = − ── x + 3 of / or
y = − ── x + 3
2
2
grafieke / graphs
y = −2x + 6
Y-afsnit / Y-intercept :
y = −2(0) + 6
y = 6
Y-afsnit / Y-intercept : (0 ; 6)
y = −2x + 6
0 = −2x + 6
x = 3
X-afsnit / X-intercept : (3 ; 0)
1
y = − ─── x + 3ASD
2
1
Y-afsnit / Y-intercept :
y = − ─── (0) + 3
2
y = 3
Y-afsnit / Y-intercept : (0 ; 3)
1
X-afsnit / X-intercept :
0 = − ─── x + 3
2
x = 6
X-afsnit / X-intercept : (6 ; 0)
4.1
f(x) = 2x2
y = 2x2 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 2y2 . . . Ruil x en y om / Swop x and y
2y2 = x
x
y2 = ──
2
y = ± √ (x / 2)
g(x) = f−1 (x) = ± √ (x / 2)
4.3 f(x) : Definisieversameling / Domain
{x | x ∈ R}
Waardeversameling / Range
{y | y ≥ 0; y ∈ R}
g(x) : Definisieversameling / Domain
{x | x ≥ 0; x ∈ R}
Waardeversameling / Range
{y | y ∈ R}
4.2 grafieke / graphs
−
4.4
f : g(x) = 2x2
y = 2x2 . . . Vervang f(x) met y /
Substitute y for f(x)
x = 2y2 . . . Ruil x en y om / Swop x and y
2y2 = x
x
y2 = ──
2
y = ± √ (x / 2)
g−1 (x) = ± √ (x / 2) ; y ≥ 0
4.6 f : Definisieversameling / Domain
{x | x ∈ R}
Waardeversameling / Range
{y | y ≥ 0; y ∈ R}
g : Definisieversameling / Domain
{x | x ≥ 0; x ∈ R}
Waardeversameling / Range
{y | y ∈ R}
4.5 grafieke / graphs
−
5.1 y = ax2
By / At P : − 3 = a(1)2
a = − 3
g(x) = − 3x2
5.2 y = − 3x2 . . . vervang g(x) met y
substitute y for f(x)
x = − 3y2 . . . Ruil x en y om / Swop x and y
− 3y2 = x
x
y2 = − ──
3
y = ± √ (x / 3)
g−1 (x) = ± √ (x / 3)
5.4 grafieke / graphs
−
5.3
g(x)
g −1 (x)
Definisieversameling / Domain : {x | x ∈ R}
{x | x ≥ 0; x ∈ R}
Waardeversameling / Range : {y | y ≥ 0; y ∈ R}
{y | y ∈ R}
6.1 f(x) = px2
By / At A : − 4 = p(−4)2
1
p = − ──
4
f(x) = − (1/4)x2
6.2 y = mx + c
By / At Y-as / Y-axis : y = mx − 3
By / At A
: −4 = m(− 4) − 3
m = (1/4)
g(x) = (1/4)x − 3
x = 12
x-afsnit / x-intercept is (12 ; 0)
6.3 grafieke / graphs
7.1 f(x) = x2
: y = x2
(x ; y) → (y ; x)
x = y2
y = ± √x
f − 1 (x) = ± √x
7.3 x ≥ 0 OF / OR
x ≤ 0
7.2 grafieke / graphs
8.1 f(x) = y = (x − 1)2 − 4
DP / TP is (1 ; 4)
Y-afsnit / Y-intercept : y = (0 − 1)2 − 4
= − 3
Y-afsnit / Y-intercept is (0 ; − 3)
X-afsnitte / X-intercepts :
(x − 1)2 − 4 = 0
(x − 1)2 = 4
(x − 1) = ± √4
(x − 1) = ± 2
x = 3 of / or x = − 1
(3 ; 0) of / or ( 1 ; 0)
8.2 E is die punt / the point (− 4 ; 1)
F is die punt / the point (0 ; 3)
G is die punt / the point (0 ; − 1)
H is die punt / the point (− 3 ; 0)
8.3 EHF
(x ≤ 1 → y ≤ 1)
9.1 Draaipunt gegee gebruik /
Turning point given use
y = a(x - p)2 + q
By / At (2 ; −4) : y = a(x − 2)2 − 4
By / At (0 ; 0) : 0 = a(0 − 2)2 − 4
0 = 4a − 4
a = 1
f(x) = (x − 2)2 − 4
= x2 − 4x + 4 − 4
= x2 − 4x
9.2 By / At B : x(x − 4) = 0
x = 0 of / or x = 4
B (4 ; 0)
9.3
f(x) is 'n een-tot-baie funksie omdat vir elke
y-waarde is daar twee
ooreenstemmende x-waardes.
f(x) is a one-to-many function because for
every y-value there are two
corresponding x-values.
9.5
9.4
Definisieversameling / Domain
Waardeversameling / Range
f(x)
{ x | x ∈ ℜ}
{ y | y ≥ − 4 ; y ∈ ℜ}
f −1 (x)
{ x | x ≥ − 4 ; x ∈ ℜ}
{ y | y ∈ ℜ}
9.6
y ≤ 2 OF / OR y ≥ 2
10.1 Draaipunt gegee gebruik /
Turning point given use
y = a(x - p)2 + q
By / At (− 1 ; 4) : y = a(x + 1)2 + 4
By / At (0 ; 3) : 3 = a(0 + 1)2 + 4
3 = a + 4
a = − 1
g(x) = − (x + 1)2 + 4
= − x2 − 2x + 3
= 3 − 2x − x2
10.2 By / At S en / and T : − x2 − 2x + 3 = 0
x2 + 2x − 3 = 0
(x + 3)(x − 1) = 0
x = − 3 of / or x = 1
S (− 3 ; 0) OF / OR T (1 ; 0)
10.3
Definisieversameling / Domain
g(x)
{ x | x ∈ ℜ}
g −1(x)
{ x | x ≤ 4 ; x ∈ ℜ}
Waardeversameling / Range
g(x)
{ y | y ≤ 4 ; y ∈ ℜ}
g −1(x)
{ y | y ∈ ℜ}
10.4
10.5 g(x) :
y = − (x + 1)2 + 4
g− 1(x) :
x = − (y + 1)2 + 4
(y + 1)2 = 4 − x
(y + 1) = ± √(4 − x)
y = ± √(4 − x) − 1
11.1
y = a√x
By / At
P(4 ; 1) : 1 = a√4
= 2a
1
a = ──
2
1
y = ── √x
2
1
11.2
x = ── √y
2
(√y)2 = (2x)2
y = 4x2
f −1(x) = 4x2
11.3 Refleksie in x-as / Reflection is x-axis
(x ; y) → (x ; − y)
y = 4x2 → − y = 4x2
g(x) = − 4x2
11.4
12.1 Draaipunt gegee gebruik /
Turning point given use
y = a(x - p)2 + q
By / At (1 ; −4) :
p = − 1; q − 4
∴ y = a(x − 1)2 − 4
By / At (2 ; −3) :
−3 = a(2 − 1)2 − 4
= a− 4
a = 1
∴ f(x) = (x − 1)2 − 4
12.2 f−1(x) :
y = (x − 1)2 − 4 → y = (x − 1)2 − 4
x = (y − 1)2 − 4
(y − 1) = √(x + 4)
y = √(x + 4) + 1
∴ f−1(x) = √(x + 4) + 1 ; y ≥ 1
X-afsnit geen X-afsnit nie, y ≥ 1 /
X-intercept no X-intercept, y ≥ 1
Y-afsnit / Y-intercept
y = √(0 + 4) + 1
= 2 + 1
= 3
Y-afsnit / Y-intercept is (0 ; 3)
12.4
12.3 g(x) : Refleksie in X-as / Reflection in X-axis
(x ; y) → (x ; −y)
∴ y = (x − 1)2 − 4
→ − y = (x − 1)2 − 4
y = − (x − 1)2 − 4
g(x) = − (x − 1)2 − 4
13.5
14.1
f(x) = 2x
y = 2x
x = 2y ruil x en y om / swop x and y
y = log2 x
f −1 (x) = log2 x
ONTHOU : As f(x) = 5x, dan kan jy sy
inverse dadelik neerskryf as
f −1 (x) = log5 x
14.3
h(x) = 5x
Skryf direk neer / Write down directly
h −1 (x) = log5 x
14.5
q(x) = 4 −x
q(x) = (1/4)x
q −1 (x) = log(1/4) x
q −1 (x) = − log4 x
14.1
f(x) = 2x
y = 2x
x = 2y ruil x en y om / swop x and y
y = log2 x
f −1 (x) = log2 x
REMEMBER : If f(x) = 5x, then you can
write down its inverse
as f −1 (x) = log5 x
14.4
p(x) = (1/2)x
Skryf direk neer / Write down directly
p −1 (x) = log(1/2) x
p −1 (x) = − log2 x
14.6
r(x) = 7 −x
r(x) = (1/7)x
r −1 (x) = log(1/7) x
r −1 (x) = − log7 x
15.1 f(x) = ax
By / At P(3 ; 8) :
8 = a3
a3 = 23
a = 2
f(x) = 2x
15.2 Definisieversameling / Domain
{x | x ∈ ℜ}
Waardeversameling / Range
{y | y > 0; y ∈ ℜ}
15.3 X-afsnit / X-intercept :
Geen X-afsnit nie want y > 0
No X-intercept because y > 0
Y-afsnit / Y-intercept is (0 ; 1)
15.4 f−1 (x) : y = log2 x
15.5 X-afsnit / X-intercept is (1 ; 0)
Y-afsnit / Y-intercept :
Geen Y-afsnit nie want x > 0
No Y-intercept because x > 0
15.6
16.1 f(x) = ax
By / At (−3 ; 27) :
27 = a−3
a−3 = (3−1)−3
a = 3−1 = 1/3
f(x) = (3−1)x = (1/3)x
16.3 Refleksie in X-as / Reflection in X-axis :
(x ; y) → (x ; −y) :
y = (3−1)x → −y = (3−1)x
y = − (3−1)x
h(x) = − (3−1)x = − (1/3)x
16.5 x ≤ 0 of / or x ≥ 0
16.2 f(x) = 3x
: y = 3x
f−1 (x) :
x = 3y
y = log3 x
f−1 (x) = log3 x
16.4 g(x) = 9x2
: y = 9x2
x = 9y2
√x
y = ± ────
3
17.1 f(x) = px
y = px
By / At A(−2 ; 25) :
25 = p−2
52 = (p−1)2
5 = p−1
p = 5−1
f(x) = (1/5)x = 5− x
17.2 Refleksie in X-as / Reflection in X-axis :
(x ; y) → (x ; −y) :
y = (5−1)x → −y = (5−1)x
y = − (5−1)x
g(x) = − (5−1)x = − (1/5)x
17.3 Refleksie in y = x / Reflection in y = x :
(x ; y) → (y ; x) :
y = (5−1)x → x = (5−1)y
y = log(1/5) x = − log5 x
h(x) = log(1/5) x = − log5 x
17.4
17.5
f(x) g(x)
Definisieversameling / Domain
{x | x ∈ ℜ}
{x | x ≥ 0; x ∈ ℜ}
Waardeversameling / Range
{y | y ≥ 0; y ∈ ℜ}
{y | y ∈ ℜ}
18.1 f(x) = logb x
y = logb x
By / At A(8 ; 3) :
3 = logb 8
b3 = 8 = 23
b = 2
f(x) = log2 x
18.2 Refleksie in X-as / Reflection in X-axis :
(x ; y) → (x ; −y) :
y = log2 x →
−y = log2 x
y = − log2 x
y = log(1/2) x
[− logb a = log(1/b) a]
g(x) = − log2 x =
log(1/2) x
18.3 Refleksie in y-as / Reflection in y-axis :
(x ; y) → (− x ; y) :
y = log2 x → y = log2 − x
h(x) = log2 (− x)
18.4