Question

How do you verify that `f(x)=3x+5; g(x)=1/3x-5/3` are inverses?

Answer 1

If `(f @ g)(x) = x` and `(g @ f)(x) = x` then `f(x) and g(x)` are inverse functions

Explanation:

There are three ways to verify that these functions are inverse functions:

1. `(f @ g)(x) = f(g(x)) = 3(1/3x-5/3)+5 = x-5+5 = x` `(g @ f)(x) = g(f(x)) = 1/3(3x+5)-5/3 = x + 5/3 - 5/3 = x` Therefore `f(x) and g(x)` are inverse functions.

2. Let `f(x) = y`. Interchange `x` with `y`, then solve for `y`. You should get `g(x)`: `x = 3y+5`, `x-5=3y`, `y=1/3x - 5/3 = g(x)`

3. Graph both functions and the `y=x` line. The functions should be reflections of each other across the `y=x` line. This means for each point `(a,b)` from `f(x)` there should be a `(b,a)` point on `g(x)`.