Question

find the inverse of the function.\\
g(x) = \sqrt{x} + 7\\
write your answer in the form a(bx + c)^2 + d, where a, b, c, and d are constants. enter the domain of the inverse in the form: x ≤ # or x ≥ #. simplify any fractions.\\
g^{-1}(x) =

Explanation:

Step1: Replace \( g(x) \) with \( y \)

We start by letting \( y = g(x) \), so the function becomes \( y = \sqrt{x} + 7 \).

Step2: Solve for \( x \) in terms of \( y \)

First, subtract 7 from both sides of the equation: \( y - 7 = \sqrt{x} \).
Then, square both sides to eliminate the square root: \( (y - 7)^2 = x \).

Step3: Replace \( x \) with \( g^{-1}(x) \) and \( y \) with \( x \)

To find the inverse function, we swap \( x \) and \( y \) and write \( g^{-1}(x) \) instead of \( y \). So we get \( g^{-1}(x) = (x - 7)^2 \).
Now, let's analyze the domain of the inverse function. The original function \( g(x)=\sqrt{x}+7 \) has a range of \( y\geq7 \) (since the square root of a non - negative number is non - negative, and then we add 7). The domain of the inverse function is the range of the original function. So the domain of \( g^{-1}(x) \) is \( x\geq7 \).

Answer:

\( g^{-1}(x)=(x - 7)^2 \), \( x\geq7 \)