Answer: No
Step-by-step explanation:
There are two ways to solve this. The first way is to calculate g(f(x)) (or f(g(x))). If f and g are inverse functions f(g(x)) = g(f(x)) = x.
g(f(x)) = (1/4)(2sqrt(x - 5))^2 - 5
= (1/4)*2^2*sqrt(x - 5)^2 - 5
= (1/4)*4*(x - 5) - 5
= x - 5 - 5
= x - 10
g(f(x)) = x - 10 =/= x, so the two functions are not inverse functions.
You can also solve this by using a test value a and calculating g(f(a)). If f and g are inverse functions, the result will be a.
For example, let a = 6.
f(6) = 2*sqrt(6 - 5)
= 2*sqrt(1)
= 2*1
= 1
g(f(6)) = g(1)
= (1/4)*1^2 - 5
= (1/4)*1 - 5
= 1/4 - 5
= -4 3/4
=/= 6
Therefore f and g are not inverse functions
Please note that this technique does not work in the other direction. If g(f(a)) = a, this does NOT prove that f and g are inverse functions.
