Help with exponent problem and inverse functions?

Question

Can some one show me how to solve the equation, 4^x + 4^-x = 5/2, algebraically?
I also need help finding the inverse fuction of sqrt(3-x), which I think is x^2 - 3, with domain restrictions. Any help is appreciated. Thanks.

Answer 1

Let m = 4^x, which lets you rewrite the equation as
m + 1/m = 5/2.
(m^2+1)/m = 5/2
2m^2 + 2 = 5m
2m^2 - 5m + 2 = 0
(2m - 1)(m - 2) = 0
2m - 1 = 0 --> m = 1/2
m - 2 = 0 --> m = 2

But m = 4^x, so 4^x = 2^(2x) = (1/2) = 2^-1 --> 2x = -1 --> x = -1/2

Also, 4^x = 2^(2x) = 2 = 2^1 --> 2x = 1 --> x = 1/2

So, the solutions are x = -1/2 and x = 1/2.

Answer 2

Unfortunately I don't think you can solve something like 4^x + 4^-x = 5/2 algebraically. Playing around with the exponents always gives you one extra term to deal with.

The inverse of √(3-x) is actually 3 - x^2