Question

What is the inverse of the function f (x) = 3(x + 5)2 – 4, such that x ≤ –5?

Answer 1

$A)f^{-1}(x)=-5+\sqrt[]{\frac{x+4}{3}}$

1) To find out the inverse function, of that one-to-one function we need to proceed with the following steps:

$f(x)=3(x+5)^2-4,f^{-1}(x)=?$

2) Swap the variables, and isolate the y variable on the left side:

$\begin{gathered} f(x)=3(x+5)^2-4 \\ y=3(x+5)^2-4 \\ x=3(y+5)^2-4 \\ -3(y+5)^2=-x-4 \\ \frac{3\mleft(y+5\mright)^2}{3}=\frac{x+4}{3} \\ (y+5)^2=\frac{x+4}{3} \end{gathered}$

Now we need to get rid of that square binomial, taking the square root on both sides:

$\begin{gathered} (y+5)^2=\frac{x+4}{3} \\ \sqrt[]{(y+5)^2}=\sqrt[]{\frac{x+4}{3}} \\ y+5=\sqrt[]{\frac{x+4}{3}} \\ y=\sqrt[]{\frac{x+4}{3}}-5 \\ f^{-1}(x)=\sqrt[]{\frac{x+4}{3}}-5 \end{gathered}$

And that is the final answer.