Page 64 - Algebra

11. f(x) = 𝑥 |𝑥|
The function is defined as f(x) = 𝑥, 𝑥≥0
–x, x<0
For 𝑥 ≥ 0,
f(x)=𝑥 =1 𝑥
For x < 0 f(x)= 𝑥 =–1
f(x) gives the output as 1 and –1. So, range = {1, –1}
12. The function is not defined for 𝑥2 Domain is R – {−√6, √6, }
Now,
y = 𝑥2−3
𝑥2−6
yx2 – 6y = x2 – 3 yx2 –x2 =6y–3
x2 (y -1) = 3(2y – 1) x2 =3(2y–1)
(y −1)
As x2 is a perfect square so,
3(2y – 1) ≥ 0 (y −1)
3(2y – 1) ≥ 0 and (y – 1) ≥ 0 y ≥ 1 and y ≥ 1, but y ≠ 1
2
So, y ≥ 1 and y > 1 2
1
Range is (-∞, 2] 𝑈 (1, ∞)
−𝑥
x ≥ –2 or x ≥ 7
− 6 = 0 or x = ±√6
13. bx is defined for all values of x, where b > 0
ó0
14. logab is defined when a > 0, b > 0 and a ≠ 1
(𝑥2 − 5𝑥 − 14)
x2 – 5x – 14 ≥ 0
x2 – 7x + 2x – 14 ≥ 0
x(x – 7) + 2(x – 7) ≥ 0
(x + 2) (x – 7) ≥ 0
–2 ≤ x ≤ 7
Therefore, (x + 2) > 0 and 15𝑥 + 8 − 2𝑥2 ≥ 0
x > –2 and 2x2 – 15x – 8 ≤ 0
x > –2 and 2x2 – 16x + x – 8 ≤ 0
x > –2 and 2x(x – 8) + 1(x – 8) ≤ 0
x > –2 and (x – 8) (2x +1) ≤ 0
x > –2 and x ≤ 8 or x ≤ –
1 2
x > –2 and –
1 2
≤ x ≤ 8
−1
or 2 < x ≤ 8
Page 63 of 177
Algebra I & II