Final answer:
To prove that f(x) and g(x) are inverses, we show that composing one with the other gives the identity function, which means f(g(x)) = x and g(f(x)) = x. After substituting and simplifying both compositions, we confirm that f(x) and g(x) are indeed inverse functions.
Step-by-step explanation:
To prove that f(x) and g(x) are inverses of each other, we must show that f(g(x)) = x and g(f(x)) = x. Let's start by finding f(g(x)).
Given f(x) = (2x+1)/3, we plug in g(x) for x:
- Let g(x) = (3x-1)/2
- Then f(g(x)) = f((3x-1)/2) = (2((3x-1)/2)+1)/3
- After simplifying, f(g(x)) = ((3x-1)+1)/3 = (3x)/3 = x
Now let's find g(f(x)).
- Plug f(x) into g(x):
- g(f(x)) = g((2x+1)/3) = (3((2x+1)/3)-1)/2
- After simplifying, g(f(x)) = ((2x+1)-1)/2 = (2x)/2 = x
Since both f(g(x)) and g(f(x)) equal x, we can conclude that f(x) and g(x) are indeed inverse functions of each other.