Type the correct answer in each box. Use numerals instead of words for numbers.

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]\( f^{-1}(x) = x^2 + \square \)[/tex], where [tex]\( x \geq \square \)[/tex].



Answer :

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].