Question

Questions 11-14
maths functions

Answer 1

11)
`\sf \left(-\infty \:,\:\infty \:\right)`

12)
`\sf x > 0` or
`\sf \left(0,\:\infty \:\right)`

13)
`\sf x < 0` or
`\sf \left(-\infty \:,\:0\right)`

14)
`\sf x\ge \:0` or
`\sf \:[0,\:\infty \:)`

Step-by-step explanation:

(11)

`f(x) > -2`

`\sf ((1)/(2) )^x - 2 > -2`

`\sf ((1)/(2) )^x > -2+2`

`\sf ((1)/(2) )^x > 0`

This statement is applicable for all value of x.

`\left(-\infty \:,\:\infty \:\right)`

(12)

`g(x) < -2`

`\sf -(2)/(x) -2 < -2`

`\sf -(2)/(x) < -2 + 2`

`\sf -(2)/(x) < 0`

`\sf (1)/(x) > 0`

`\sf x > 0`

(13)

`f(x) > -1`

`\sf ((1)/(2) )^x - 2 > -1`

`\sf ((1)/(2) )^x > -1+2`

`\sf ((1)/(2) )^x > 1`

`\sf ln(((1)/(2) )^x) > ln(1)`

`\sf xln((1)/(2) ) > ln(1)`

`\sf x < ln(1)/ ln((1)/(2) )`

`\sf x < 0`

(14)

`-2 < f(x) < -1`

1st

`\sf -2 < ((1)/(2) )^x - 2`

`\sf 0 < ((1)/(2) )^x`

True for all x

2nd

`\sf ((1)/(2) )^x - 2 \leq -1`

`\sf ((1)/(2) )^x \leq 1`

`\sf x \geq 0`

So in total,

`\sf x\ge \:0`

These solution's can be found by determining them graphically.