Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$?
I want to find the value of $x$ that makes $g(x)=15$. I know that $g(x)=3f^{-1}(x)$, so I can substitute that into the equation and get $3f^{-1}(x)=15$. To isolate $f^{-1}(x)$, I can divide both sides by $3$ and get $f^{-1}(x)=5$. Now I need to figure out what $f^{-1}(x)$ means. It is the inverse function of $f(x)$, which means that it undoes what $f(x)$ does. So if $f(x)=\frac{24}{x+3}$, then $f^{-1}(x)$ is the value of $y$ that makes $\frac{24}{y+3}=x$. To find this value of $y$, I can cross-multiply and get $24=x(y+3)$. Expanding the brackets, I get $24=xy+3x$. Subtracting $3x$ from both sides, I get $24-3x=xy$. Dividing both sides by $x$, I get $\frac{24-3x}{x}=y$. This is the expression for $f^{-1}(x)$. Since I want $f^{-1}(x)=5$, I can set this expression equal to $5$ and solve for $x$. I get $\frac{24-3x}{x}=5$. Multiplying both sides by $x$, I get $24-3x=5x$. Adding $3x$ to both sides, I get $24=8x$. Dividing both sides by $8$, I get $x=3$. This is the value of $x$ that makes $g(x)=15$. 