Question: a) Given the functions $f(x) = x + 2$ and $g(x) = 3^x$, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
b) Determine all values for $x$ for which $f(g(x)) = g(f(x))$.
*For part a), I got the equation:
$(f ∘ g)(x) = 3^x + 2$
and
$(g ∘ f)(x) = 3^{x+2}$
But I don't know if it's right or not. I'd appreciate if anyone can tell me if I got it right or not. And if not, what I did wrong. I believe it's right because I used the simplified formula of $f(g(x))$ and $g(f(x))$. But still, I just want to make sure that I'm correct.
For part c.) I don't know how to solve the question at all. Here's what I started with:
$3^x + 2 = 3^{x+2}$
But I don't know how to continue on. I don't know how to get rid of the 2. I'm trying to figure out a way where I can get all the bases to be the same so I can solve for x. But I don't think it's going to work like that and I don't know what else to do. I'd be thankful if anyone can help me out!
Your answer for the first pat is correct. For solving $3^{x}+2=3^{x+2}$ write this as $3^{x} (3^{2}-1)=2$ or $3^{x}=\frac 1 4$. Take logarithm to finish.