Question

find the inverse of the function.
( h(x) = sqrt{x - 3} )
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 leq # ) or ( x geq # ). simplify any fractions.
( h^{-1}(x) = )
, ( x )

Explanation:

Step1: Let \( y = h(x) = \sqrt{x - 3} \)

We start by letting \( y \) equal the function \( h(x) \). This is the first step in finding the inverse function, as we will later swap \( x \) and \( y \).

Step2: Swap \( x \) and \( y \)

We get \( x = \sqrt{y - 3} \). Swapping \( x \) and \( y \) is a key step in finding the inverse of a function, as it reflects the graph over the line \( y = x \).

Step3: Solve for \( y \)

First, square both sides to eliminate the square root: \( x^2 = y - 3 \). Then, add 3 to both sides to solve for \( y \): \( y = x^2 + 3 \). So the inverse function \( h^{-1}(x) = x^2 + 3 \).

Step4: Find the domain of the inverse function

The original function \( h(x) = \sqrt{x - 3} \) has a range of \( y \geq 0 \) (since the square root of a non - negative number is non - negative). The domain of the inverse function is the range of the original function. So for \( h^{-1}(x)=x^2 + 3 \), the domain is \( x\geq0 \).

Answer:

The inverse function \( h^{-1}(x)=\boldsymbol{x^2 + 3} \) and the domain of the inverse function is \( x\geq\boldsymbol{0} \) (in the form \( x\geq\# \), here \(\# = 0\)).