To determine the inverse of the given function, follow these steps:
1. Start with the given function:
[tex]\[
f(x) = \sqrt{x - 4}
\][/tex]
2. Change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt{x - 4}
\][/tex]
3. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt{y - 4}
\][/tex]
4. Solve for [tex]\( y \)[/tex]. Square both sides to remove the square root:
[tex]\[
x^2 = y - 4
\][/tex]
5. Isolate [tex]\( y \)[/tex] by adding 4 to both sides:
[tex]\[
y = x^2 + 4
\][/tex]
Therefore, the inverse function can be written as:
[tex]\[
f^{-1}(x) = x^2 + 4
\][/tex]
Since the original function [tex]\( f(x) = \sqrt{x - 4} \)[/tex] is only defined for [tex]\( x \geq 4 \)[/tex], the inverse function should also respect this domain. For the inverse function, which is [tex]\( f^{-1}(x) = x^2 + 4 \)[/tex], the corresponding value for [tex]\( x \)[/tex] should be adjusted to the domain of the original function:
- The constant term added to [tex]\( x^2 \)[/tex] is [tex]\( 4 \)[/tex].
- The range limit for the original function is [tex]\( 4 \)[/tex], meaning the inverse function is valid for [tex]\( x \geq 4 \)[/tex].
Thus, filling in the boxes correctly:
Consider this function:
[tex]\[
f(x)=\sqrt{x-4}
\][/tex]
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]\( x \geq 4 \)[/tex].
