Let's use the inverse composition rule to verify that f(x) = x - 2/5 and g(x) = 5x + 2 are indeed inverse functions.
f(g(x)) g(f(x))
f(g(x)) = g(x) - 2/5 g(f(x)) = (5)(f(x)) + 2
f(g(x)) = (5x + 2) - 2/5 g(f(x)) = (5)(x - 2/5) + 2
f(g(x)) = 5x + 2 - 2/5 g(f(x)) = 5x + 2 + 2
f(g(x)) = 5x - 8/10 = 5x + 4/5 g(f(x)) = 5x + 4
f(g(x)) = 5x + 4/5
Thus, functions f(x) and g(x) are not inverses because f(g(x)) = 5x + 4/5 and g(f(x)) = 5x + 4, f(g(x)) and g(f(x)) must be = x to be called inverses of each other.
If, f(x) = (x-2)/5 and g(x) = 5x + 2
f(g(x)) g(f(x))
f(g(x)) = (g(x) - 2)/5 g(f(x)) = (5)(f(x)) + 2
f(g(x)) = ((5x + 2) - 2)/5 g(f(x)) = (5)((x - 2)/5) + 2
f(g(x)) = (5x + 2 - 2)/5 g(f(x)) = x - 2 + 2 = x
f(g(x)) = 5x/5 = x g(f(x)) = x
f(g(x)) = x
Thus, functions f(x) and g(x) are inverses because f(g(x)) = x and g(f(x)) = x, f(g(x)) and g(f(x)) must be = x to be called inverses of each other.