Memo: Topic 2 Inverse functions
1.1.1
True, interchange x and y
1.1.2
True, more than one x value are mirrored on one y value
1.1.3
False, the inverse will have two y values for each x-value
1.1.4
False, D = {-2, 2; 4; 6}
1.1.5
False, reflection in the line y = x
1.2.1
𝑓 −1 : 𝑥 = 𝑦 2
1.2.2
Not a function, the inverse is a one: many relationship, for each x there are two y values.
1.2.3
𝑥 ≤ 0, 𝑥 ∈ 𝑅 ( 𝑜𝑟 𝑥 ≥ 0, 𝑥 ∈ 𝑅)
1
2
∴ 𝑓 −1 (𝑥) = ±√2𝑥 ;
1.2.4
1.2.5
1 2
𝑥
2
= ±√2𝑥
𝑥 4 − 8𝑥 = 0 (𝑠𝑞𝑢𝑎𝑟𝑒 𝑎𝑛𝑑 × 4)
𝑥(𝑥 3 − 8) = 0 ∴ 𝑥 = 0 𝑜𝑓 𝑥 = 2
𝑦 = 0 0𝑓 𝑦 = 2 (y > 0 for f)
Points of intersection (0;0) and (2;2)
𝑥≥0
2
3
3
2
1.3.1
𝑓 −1 : 𝑥 = 𝑦
∴ 𝑓 −1 (𝑥) = 𝑥
1.3.2
𝑔−1 : 𝑥 = −3𝑥 − 9
1.4
3
𝑥
2
1.5.1
𝑓 −1 (𝑥) = ±√𝑥 ;
1
3
=− 𝑥−3
1
∴ 𝑔−1 (𝑥) = − 3 𝑥 − 3
∴𝑥=−
18
11
𝑎𝑛𝑑 𝑦 = −
27
11
𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 (−
𝑥≥0
1.5.2
1.5.3
For each x-value in the inverse there are two y-values – not a function
1.5.4
𝑥 ≤ 0, 𝑥 ∈ 𝑅 ( 𝑜𝑓 𝑥 ≥ 0, 𝑥 ∈ 𝑅)
1.6.1
1
16
1.6.2
= 𝑎2
1
∴𝑎=4
18
27
;− )
11
11
2.1.1
3 = 𝑘1
∴𝑘=3
𝑝(𝑥) = 𝑎(𝑥 − 3)2 + 1
∴ 3 = 𝑎(1 − 3)2 + 1
1
∴𝑎=2
1
1
∴ 𝑦 = 𝑥 2 − 3𝑥 + 5
2
2
𝑏 = −3 𝑎𝑛𝑑 𝑐 =
1
52
𝑔 ∶ 1 = 𝑙𝑜𝑔𝑚 3
.
y
p(x)
f(x)
.Q(1 ; 3)
G.
.E
.H .
F
∴𝑚=3
𝐺𝐹 = √12 + 12 = √2 = 1.41
2.1.2
𝐸𝐹 = 1,
2.1.3
𝑓 −1 (𝑥) = 𝑙𝑜𝑔3 𝑥 𝑎𝑛𝑑 𝑔−1 (𝑥) = 3𝑥
2.1.4
𝑓 −1 (𝑥) = 𝑙𝑜𝑔3 𝑥 = 𝑔(𝑥) 𝑎𝑛𝑑 𝑔−1 (𝑥) = 3𝑥 = 𝑓(𝑥) hence 𝑓(𝑥) 𝑎𝑛𝑑 𝑔(𝑥) are inverse
functions of each other and therefor a reflection in the line y = x ; the x and y values
interchanged.
2.2.1
𝑓(𝑥) = 𝑎 𝑥 𝑆𝑢𝑏𝑠𝑡 𝑖𝑛 ( 1; 4) ∴ 4 = 𝑎1
2.2.2
𝑃 = (0; 1) 𝑎𝑛𝑑 𝑄 = (1; 0)
2.2.3
𝑔(𝑥) = ( ) ;
1 𝑥
4
𝑎=4
𝑔−1 (𝑥) = 𝑙𝑜𝑔1 𝑥
ℎ(𝑥) = 𝑙𝑜𝑔4 𝑥 ;
4
2.3.1
Inverse is not a function because for each x value there are two y-values.
2.3.2
𝑓 −1 : 𝑥 = −2𝑦 2
2.3.3. 𝐷𝑉 = { −∞ < 𝑥 ≤ 0, 𝑥 ∈ 𝑅}
2.3.4
𝑥
∴ 𝑦 = −√− 2 ; 𝑥 ∈ (−∞; 0]
( of 𝑥 ≤ 0, 𝑥 ∈ 𝑅
𝑥
∴ 𝑓 −1 (𝑥) = −√− 2 ; 𝑥 ∈ (−∞; 0]
𝑜𝑟 𝑥 ∈ (−∞; 0])
g(x
)
x