Final answer:
After computing f(g(x)) and g(f(x)), it is evident that neither composition yields x. Therefore, f(x) and g(x) are not inverses of each other, and the correct answer is b) No, they are not inverses.
Step-by-step explanation:
To determine whether f(x) and g(x) are inverses, we need to check if f(g(x)) equals x and if g(f(x)) equals x. Given f(x) = \frac{4}{5}x + 1 and g(x) = 5x - \frac{5}{4}, let's compute f(g(x)):
f(g(x)) = f(5x - \frac{5}{4})
= \frac{4}{5}(5x - \frac{5}{4}) + 1
= 4x - 1 + 1
= 4x
Next, let's compute g(f(x)):
g(f(x)) = g(\frac{4}{5}x + 1)
= 5(\frac{4}{5}x + 1) - \frac{5}{4}
= 4x + 5 - \frac{5}{4}
= 4x + 4\frac{15}{4} - \frac{5}{4}
= 4x + \frac{20}{4}
= 4x + 5
As we see, f(g(x)) does not equal x, and g(f(x)) does not equal x; hence, f(x) and g(x) are not inverses of each other. The correct answer is b) No, they are not inverses.