To determine which function can take the value of [tex]\(-4\)[/tex], we'll analyze each function and see if setting [tex]\( y = -4 \)[/tex] yields a valid solution for [tex]\( x \)[/tex].
1. Function: [tex]\(y = \sqrt{x} - 5\)[/tex]
- To find out if [tex]\( -4 \)[/tex] is in the range:
[tex]\[
y = -4 \implies -4 = \sqrt{x} - 5
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-4 + 5 = \sqrt{x} \implies 1 = \sqrt{x} \implies x = 1
\][/tex]
- Since [tex]\( x = 1 \)[/tex] is a valid number, [tex]\( y = \sqrt{x} - 5 \)[/tex] can indeed be [tex]\(-4\)[/tex]. Thus, [tex]\(-4\)[/tex] is in the range of [tex]\( y = \sqrt{x} - 5 \)[/tex].
2. Function: [tex]\(y = \sqrt{x} + 5\)[/tex]
- To find out if [tex]\( -4 \)[/tex] is in the range:
[tex]\[
y = -4 \implies -4 = \sqrt{x} + 5
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-4 - 5 = \sqrt{x} \implies -9 = \sqrt{x}
\][/tex]
- Since the square root of a number is always non-negative, [tex]\(\sqrt{x} = -9\)[/tex] is impossible. Therefore, [tex]\(-4\)[/tex] is not in the range of [tex]\( y = \sqrt{x} + 5 \)[/tex].
3. Function: [tex]\(y = \sqrt{x+5}\)[/tex]
- To find out if [tex]\( -4 \)[/tex] is in the range:
[tex]\[
y = -4 \implies -4 = \sqrt{x+5}
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-4 = \sqrt{x + 5}
\][/tex]
- As the square root function returns a non-negative value, [tex]\(-4 = \sqrt{x + 5}\)[/tex] is impossible. Therefore, [tex]\(-4\)[/tex] is not in the range of [tex]\( y = \sqrt{x+5} \)[/tex].
4. Function: [tex]\(y = \sqrt{x-5}\)[/tex]
- To find out if [tex]\( -4 \)[/tex] is in the range:
[tex]\[
y = -4 \implies -4 = \sqrt{x-5}
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-4 = \sqrt{x - 5}
\][/tex]
- Similarly, since the square root function returns a non-negative value, [tex]\(-4 = \sqrt{x - 5}\)[/tex] is impossible. Therefore, [tex]\(-4\)[/tex] is not in the range of [tex]\( y = \sqrt{x-5} \)[/tex].
After analyzing each function, we determine that the function [tex]\( y = \sqrt{x} - 5 \)[/tex] is the only one whose range includes [tex]\(-4\)[/tex].
