Questions 1-5 please! maths functions

Question

asked Sep 1, 2023 209k views

1 Answer

Lballes · Answer 1 · 2023-09-07T04:51:38+0000

Answer:

1) x < -1

2) x ≥ -1

3) x ≤ -1

4) x < -1 and x > 0

5) -1 < x < 0

Step-by-step explanation:

(1) f(x) > 0

$\sf \rightarrow \left((1)/(2)\right)^x-2 > \:0$

$\sf \rightarrow \left((1)/(2)\right)^x > \:2$

$\sf \rightarrow \left((1)/(2)\right)^x > \:2^(-1)$

$\sf \rightarrow x < -1$

(2) f(x) ≤ 0

$\sf \rightarrow \left((1)/(2)\right)^x-2 \leq \:0$

$\sf \rightarrow \left((1)/(2)\right)^x \leq \:2$

$\sf \rightarrow \left((1)/(2)\right)^x \leq \:2^(-1)$

$\sf \rightarrow x\geq -1$

(3) g(x) ≥ 0

$\sf \rightarrow -(2)/(x)-2 \:\geq \:0$

$\sf \rightarrow -(2)/(x) \:\geq \:2$

$\sf \rightarrow (1)/(x) \:\geq \:-1$

$\sf \rightarrow x \: \leq \:-1$

(4) g(x) < 0

$\sf \rightarrow -(2)/(x)-2 \: < \: 0$

$\sf \rightarrow x < -1 \ or \ x > 0$

Observe the graphs a find solution for g(x) < 0

(5) f(x) < g(x)

$\sf \rightarrow \left((1)/(2)\right)^x-2 < -(2)/(x)-2$

$\sf \rightarrow -1 < x < 0$