We need to determine which of the given functions has a range that includes -4:
1. [tex]\( y = \sqrt{x} - 5 \)[/tex]
2. [tex]\( y = \sqrt{x} + 5 \)[/tex]
3. [tex]\( y = \sqrt{x+5} \)[/tex]
4. [tex]\( y = \sqrt{x-5} \)[/tex]
We are given [tex]\(x = 59.0\)[/tex]. Let's compute the values for each function and see where -4 falls within the range.
First, calculate [tex]\( \sqrt{59} \)[/tex]:
[tex]\[ \sqrt{59} \approx 7.681145747868608 \][/tex]
Using this value, compute:
1. [tex]\( y = \sqrt{59} - 5 \)[/tex]
[tex]\[
y = 7.681145747868608 - 5 = 2.681145747868608
\][/tex]
The result is approximately [tex]\(2.681\)[/tex].
2. [tex]\( y = \sqrt{59} + 5 \)[/tex]
[tex]\[
y = 7.681145747868608 + 5 = 12.681145747868609
\][/tex]
The result is approximately [tex]\(12.681\)[/tex].
For the next two functions, we need to adjust the argument inside the square root:
3. [tex]\( y = \sqrt{59 + 5} = \sqrt{64} \)[/tex]
[tex]\[
y = \sqrt{64} = 8
\][/tex]
The result is [tex]\(8\)[/tex].
4. [tex]\( y = \sqrt{59 - 5} = \sqrt{54} \)[/tex]
[tex]\[
y = \sqrt{54} \approx 7.3484692283495345
\][/tex]
The result is approximately [tex]\(7.348\)[/tex].
Now, let's look at the ranges of these functions to see if -4 is included:
1. For [tex]\( y = \sqrt{59} - 5 \)[/tex], the range is given as approximately [tex]\(2.681\)[/tex], which does not include -4.
2. For [tex]\( y = \sqrt{59} + 5 \)[/tex], the range is given as approximately [tex]\(12.681\)[/tex], which does not include -4.
3. For [tex]\( y = \sqrt{x+5} \)[/tex], the result is [tex]\(8\)[/tex], which does not include -4.
4. For [tex]\( y = \sqrt{x-5} \)[/tex], the result is approximately [tex]\(7.348\)[/tex], which does not include -4.
Review of the functions confirms that only [tex]\( y = \sqrt{x} - 5 \)[/tex] gives a result that includes -4 in its range. Therefore, the function whose range includes -4 is:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
