Determine whether f(x)= 5x+1/x and g(x)=x/5x+1 are inverse functions. Explain how you know

Answer 1

To determine if f(x) and g(x) are inverse functions, we need to check if their composite function is equal to the identity function.

First, we find the composite function f(g(x)):
f(g(x)) = f(x/(5x+1))
= 5(x/(5x+1)) + 1/(x/(5x+1))
= (5x)/(5x+1) + (5x+1)/x
= (5x^2 + x)/(5x+1) + (5x+1)/x
= (5x^2 + x + (5x+1)(5x+1))/(x(5x+1))
= (5x^2 + x + 25x^2 + 10x + 1)/(x(5x+1))
= (30x^2 + 11x + 1)/(x(5x+1))

Next, we find the composite function g(f(x)):
g(f(x)) = g(5x + 1/x)
= (5x + 1/x)/(5(5x + 1/x) + 1)
= (5x + 1/x)/(25x + 5/x + 1)
= (5x^2 + 1)/(25x^2 + 5 + x)
= (5x^2 + 1)/(25x^2 + x + 5)

Comparing f(g(x)) and g(f(x)), we see that they are not equal. Therefore, f(x) and g(x) are not inverse functions.