Answer:
Inverse function: [tex]y=\sqrt{x+4}[/tex]
Step-by-step explanation:
- An inverse function [tex]f^{-1}[/tex] is a function such [tex]f^{-1}(f(x))=x[/tex], for all x.
- In practice, to find the inverse function, we just have to swich "x" by "y" in the original function, and clear y, which would be the inverse funtion.
- In this case, [tex]y=f(x)=x^2-4[/tex]. Then, if we swich x by y, we have the following expression: [tex]x=y^2-4[/tex]. Now, we just have to clear y, as a function of x, by doing the following:
- Add 2 both sides of the equation. This would yield [tex]x+4=y^2[/tex].
- Take square root both sides of the equation. This would yield [tex]y=\sqrt{x+4}[/tex]. Then we have the inverse function!
- To verify the process, we replace f(x) (our original expression) into the new equation [tex]f^{-1}(f(x))=x[/tex], and check that equals x: [tex]f^{-1}[/tex] is [tex]f^{-1}(f(x))=\sqrt{(x^2-4)+4} =\sqrt{x^2} =x[/tex]
