To determine the inverse of the given function, we will follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt{x - 4}
\][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt{y - 4}
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
a. Square both sides of the equation to remove the square root:
[tex]\[
x^2 = y - 4
\][/tex]
b. Add 4 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
x^2 + 4 = y
\][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = x^2 + 4
\][/tex]
Next, we consider the domains of the functions. The original function [tex]\( f(x) \)[/tex] is defined for [tex]\( x \geq 4 \)[/tex] because we are taking the square root of [tex]\( (x - 4) \)[/tex]. This means that [tex]\( y \)[/tex] must be at least 4.
Since [tex]\( y \geq 4 \)[/tex] translates to [tex]\( f^{-1}(x) \)[/tex] being defined for [tex]\( x \geq 0 \)[/tex], we have the following completed answer:
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].
