Let $f(x) = 3x-8$ and $g(f(x)) = 2x^2 + 5x - 3.$ Find $g(-5).$
I want to find $g(-5)$, but I don't have a formula for $g(x)$ directly. So I need to figure out what value of $x$ makes $f(x) = -5$. I can do this by solving the equation $f(x) = -5$ for $x$. This means $3x - 8 = -5$, so $3x = 3$ and $x = 1$. Therefore, $g(-5) = g(f(1))$. Now I can plug in $x = 1$ into the formula for $g(f(x))$ and simplify. This gives me $g(f(1)) = 2(1)^2 + 5(1) - 3 = 2 + 5 - 3 = 4$. So the final answer is $g(-5) = 4$. 