Determining f -1 (x) of functions
You write the inverse of \(f(x)\) as \({f^{ - 1}}(x)\). This reverses the process of \(f(x)\) and takes you back to your original values.
Example
If \(f(x) = 7x - 2\), find \({f^{ - 1}}(x)\)
First, rearrange in terms of \(x\):
\(y = 7x - 2\)
\(7x - 2 = y\)
\(7x = y + 2\)
Remember to change \(y\) back to \(x\) when you're writing your answer:
\(x = \frac{{y + 2}}{7}\)
Thus \({f^{ - 1}}(x) = \frac{{x + 2}}{7}\)
Now use the same method to solve this question.
Question
If \(g(x) = {x^2} + 4,\,x \ge 0\) find \({g^{ - 1}}(x)\)
First, rearrange in terms of \(x\):
\(y = {x^2} + 4\)
\({x^2} = y - 4\)
\(x = \sqrt {y - 4}\)
Thus \({g^{ - 1}}(x)=\sqrt{x - 4}\)
