Can anyone explain how this math (function) problem is even possible

Look at the boundaries of your equations, and compare to the given value of x.

If x is greater than -2, and less than or equal to 5, then use the first equation to solve for f(x).

If x is greater than 5, and less than or equal to 6, use the second equation.

Since the value of x you were given is 5, use the first equation.
f(x) = 2x - 1
f(5) = 2(5) - 1
f(5) = 9

The reason you don't use the second equation is because x=5 isn't within that boundary.

"it specifically states that in order for "9" to be the correct answer to function problem, the answer (9) must be a number "LESS than or equal to -2,"
Sorry, no.
It's *not* the function f(x) which must must between -2 and 5 inclusive.
It's x which must be conform to those limits.
And as x = 5 (you're asked for f(5)), the first definition ie 2*x - 1 = 9 is the one you must use.

You are NOT LOOKING at the domains...

when x: - 2 ≤ x ≤ 5
then the function is: f(x) = 2x - 1

for x > 5 [and so on...], the function is: f(x) = x^3 - 5

Your problem is that you appear to believe that when x = 5, it is both EQUAL TO

AND GREATER THAN 5

That is NOT POSSIBLE !

f(5) = 2(5) - 1 = 9

both RIGHT and rather SIMPLE !

QED

The inequality specifies x values that are valid, not the range of the function itself. Remember x is what you put in, f(x) is what you get out.

On a graph, from -2 to 5 it would look like a line, on 5 to 6 it would look like a cubic equation.