Given the function f(x)=1-4^(2-x)[(1/2)^(x+3)], find the value/s of x such that f(x)<h(x) if h(x)=8^x

Question

Hello! I would greatly appreciate any assistance in solving this problem involving exponential functions. Thank you in advance!

Answer 1

1 - 4^2-x(1/2^{x + 3}) < 8^x; 1 - 2^{4 - 2x - x - 3} < 2^3x; 1 - 2^{1 - 3x} < 2^3x; Let u = 2^3x > 0; then we have 1 - 2/u < u;

u² - u + 2 > 0 and u > 0; So u > 0, 2^3x > 0; x is any real number: (-∞, ∞)