To determine which function's range includes [tex]\( y = 4 \)[/tex], we need to solve for [tex]\( x \)[/tex] in each equation and check if our solutions are valid within the domain of the respective functions.
1. First function: [tex]\( y = \sqrt{x} - 5 \)[/tex]
Let's set [tex]\( y = 4 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
4 = \sqrt{x} - 5
\][/tex]
Adding 5 to both sides:
[tex]\[
4 + 5 = \sqrt{x}
\][/tex]
[tex]\[
9 = \sqrt{x}
\][/tex]
Squaring both sides to eliminate the square root:
[tex]\[
9^2 = x
\][/tex]
[tex]\[
x = 81
\][/tex]
Validate whether [tex]\( x = 81 \)[/tex] is within the domain:
- The domain of [tex]\( y = \sqrt{x} - 5 \)[/tex] is [tex]\( x \geq 0 \)[/tex], which includes [tex]\( x = 81 \)[/tex]. Hence, [tex]\( y = 4 \)[/tex] is in the range of [tex]\( y = \sqrt{x} - 5 \)[/tex].
2. Second function: [tex]\( y = \sqrt{x+5} \)[/tex]
Again, let's set [tex]\( y = 4 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
4 = \sqrt{x+5}
\][/tex]
Squaring both sides to eliminate the square root:
[tex]\[
4^2 = x + 5
\][/tex]
[tex]\[
16 = x + 5
\][/tex]
Subtracting 5 from both sides:
[tex]\[
16 - 5 = x
\][/tex]
[tex]\[
x = 11
\][/tex]
Validate whether [tex]\( x = 11 \)[/tex] is within the domain:
- The domain of [tex]\( y = \sqrt{x+5} \)[/tex] is [tex]\( x \geq -5 \)[/tex], which includes [tex]\( x = 11 \)[/tex]. Hence, [tex]\( y = 4 \)[/tex] is in the range of [tex]\( y = \sqrt{x+5} \)[/tex].
3. Third function: [tex]\( y = \sqrt{x+5} \)[/tex]
This function is identical to the second function, so it will produce the same result. Solving it would follow the same steps as above and again yield [tex]\( x = 11 \)[/tex], which is valid within the function's domain.
Thus, the functions whose range includes [tex]\( y = 4 \)[/tex] are:
[tex]\[ y = \sqrt{x} - 5 \][/tex]
[tex]\[ y = \sqrt{x+5} \][/tex]
In conclusion, the function numbers are [tex]\( 1 \)[/tex] and [tex]\( 2 \)[/tex].
