Find the values of x such that f(x)<h(x) given f(x)=1-4^(2-x)[(1/2)^(x+3)] and h(x)=8^x

Question

Find the values of "x" given:

1. f(x)<h(x)
2. f(x)=1-4^(2-x)[(1/2)^(x+3)]
3. h(x)=8^x

Answer 1

Change the bases of the exponents

4 = 2²

8 = 2³

1/2 = 2^-1

f(x) becomes 1 - 2^2(2-x)2^-(x+3) = 1 - 2^4-2x2^-x-3 = 1 - 2^1-3x = 1 - 2¹ - 2^3x = -1 - 2^3x

h(x) becomes 2^3x

So, 2^3x > -1 - 2^3x

Let u = 2^3x

Therefore,

u > -1 - u

2u > -1

u> -1/2

Plug 2^3x back in

2^3x > -1/2

This occurs for all real numbers so h(x)> f(x) always