Question

What is the range of the function `f(x) = 5^x+5^(-x)+7` ?

Answer 1

`[9, oo)`

Explanation:

Given:

`f(x) = 5^x+5^(-x)+7`

Note that for any real value of `x`, we have `5^x > 0` and `5^(-x) > 0`, so `f(x) > 0`.

In addition note that `f(x)` is an even function. That is:

`f(-x) = 5^(-x)+5^x + 7 = 5^x+5^(-x)+7 = f(x)`

We find:

`f(0) = 5^0+5^0+7 = 1+1+7 = 9`

Given that `5^x` is monotonically increasing with `x` and quite drastically so (faster than any polynomial), the sum `5^x+5^(-x)` is dominated by `5^x` when `x > 0` and by `5^(-x)` when `x < 0`.

Hence the value at `x=0` is the minimum value and the range of `f(x)` is `[9, oo)`

graph{5^x+5^(-x)+7 [-5.6, 5.6, -10, 50]}