[tex]f^{-1}(x)=(x+8)^2\text{ ; }x\text{ }\ge-8\text{ (option C)}[/tex]
Explanation:[tex]f(x)\text{ = }\sqrt[]{x}\text{ - 8}[/tex]
let y = f(x)
[tex]y\text{ = }\sqrt[]{x}\text{ - 8}[/tex]
To get the inverse of f(x): first, we will interchange y and x. Then we will solve for y.
[tex]\begin{gathered} In\text{tercahnge y and x:} \\ x\text{ = }\sqrt[]{y}\text{ - 8} \end{gathered}[/tex]
Add 8 to both sides:
[tex]\begin{gathered} x\text{ + 8 = }\sqrt[]{y}\text{ - 8 + 8} \\ x\text{ + 8 = = }\sqrt[]{y} \\ \text{square both sides:} \\ (x+8)^2\text{ = (}\sqrt[]{y})^2 \\ y\text{ = }(x+8)^2\text{ } \\ \\ \text{Hence, }f^{-1}(x)\text{ = }(x+8)^2\text{ } \end{gathered}[/tex]
Domain are the inputs of a function. They are the x values.
The inverse of f(x) doesn't have a denominator.
The domain of an inverse functon is the range of the original function
Range (y values ) of original function was x ≥ -8
Hence, domain of this function is x ≥ -8
[tex]f^{-1}(x)=(x+8)^2\text{ ; }x\text{ }\ge-8\text{ (option C)}[/tex]
