Sure, let's work through the steps to determine the inverse of the given function [tex]\( f(x) = \sqrt{x-4} \)[/tex].
1. Start by changing [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{x - 4} \][/tex]
2. To find the inverse function, switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \sqrt{y - 4} \][/tex]
3. Now solve for [tex]\( y \)[/tex]:
- Square both sides to eliminate the square root:
[tex]\[ x^2 = y - 4 \][/tex]
- Add 4 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = x^2 + 4 \][/tex]
4. Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] can be written as:
[tex]\[ f^{-1}(x) = x^2 + 4 \][/tex]
5. Next, we need to determine the domain of the inverse function. The original function [tex]\( f(x) \)[/tex] has a domain of [tex]\( x \geq 4 \)[/tex] because you cannot take the square root of a negative number. For the inverse function, we consider that [tex]\( x \geq 0 \)[/tex].
Putting it all together, we have:
[tex]\[
f^{-1}(x) = x^2 + 4, \quad \text{where } x \geq 0
\][/tex]
So, for the boxes, we fill in:
To determine the inverse of the given function, change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex], switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and solve for [tex]\( y \)[/tex].
The resulting function can be written as [tex]\( f^{-1}(x) = x^2 + 4 \)[/tex], where [tex]\( x \geq 0 \)[/tex].
