Question

Questions 1-5 please!
maths functions

Answer 1

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`