Define the function $f(x) = 2x - 5$. For what value of $x$ is $f(x)$ equal to $f^{-1}(x)$?
I need to find the inverse function of $f(x)$ first. To do that, I can swap $x$ and $y$ and solve for $y$. So I get $x = 2y - 5$, and I need to isolate $y$. I can add $5$ to both sides and get $x + 5 = 2y$. Then I can divide both sides by $2$ and get $y = \frac{x + 5}{2}$. This is the inverse function, $f^{-1}(x) = \frac{x + 5}{2}$. Now I want to find the value of $x$ that makes $f(x) = f^{-1}(x)$. I can set them equal to each other and get $2x - 5 = \frac{x + 5}{2}$. I can multiply both sides by $2$ to get rid of the fraction and get $4x - 10 = x + 5$. I can subtract $x$ from both sides and get $3x - 10 = 5$. I can add $10$ to both sides and get $3x = 15$. I can divide both sides by $3$ and get $x = 5$. This is the value of $x$ that satisfies the equation. 