To find the function that has a domain of [tex]\( x \geq 5 \)[/tex] and a range of [tex]\( y \leq 3 \)[/tex], we'll analyze each given function separately:
### 1. [tex]\( y = \sqrt{x-5} + 3 \)[/tex]
Domain:
To ensure the expression under the square root is non-negative, we need:
[tex]\[ x - 5 \geq 0 \][/tex]
[tex]\[ x \geq 5 \][/tex]
Hence, the domain is [tex]\( x \geq 5 \)[/tex].
Range:
For [tex]\( x \geq 5 \)[/tex], [tex]\( \sqrt{x - 5} \)[/tex] will be non-negative (i.e., [tex]\( \sqrt{x-5} \geq 0 \)[/tex]). Consequently,
[tex]\[ y = \sqrt{x - 5} + 3 \geq 0 + 3 = 3 \][/tex]
Thus, the range is [tex]\( y \geq 3 \)[/tex].
This does not fit our required range of [tex]\( y \leq 3 \)[/tex].
### 2. [tex]\( y = \sqrt{x+5} - 3 \)[/tex]
Domain:
To ensure the expression under the square root is non-negative, we need:
[tex]\[ x + 5 \geq 0 \][/tex]
[tex]\[ x \geq -5 \][/tex]
Hence, the domain is [tex]\( x \geq -5 \)[/tex].
Range:
For [tex]\( x \geq -5 \)[/tex]:
[tex]\[ \sqrt{x + 5} \geq 0 \][/tex]
Thus,
[tex]\[ y = \sqrt{x + 5} - 3 \geq 0 - 3 = -3 \][/tex]
So the range will be [tex]\( y \geq -3 \)[/tex].
This does not fit our required range of [tex]\( y \leq 3 \)[/tex].
### 3. [tex]\( y = -\sqrt{x-5} + 3 \)[/tex]
Domain:
To ensure the expression under the square root is non-negative, we need:
[tex]\[ x - 5 \geq 0 \][/tex]
[tex]\[ x \geq 5 \][/tex]
Hence, the domain is [tex]\( x \geq 5 \)[/tex].
Range:
For [tex]\( x \geq 5 \)[/tex]:
[tex]\[ \sqrt{x - 5} \geq 0 \][/tex]
Thus,
[tex]\[ y = -\sqrt{x - 5} + 3 \leq 0 + 3 = 3 \][/tex]
So the range will be [tex]\( y \leq 3 \)[/tex].
This fits our required range of [tex]\( y \leq 3 \)[/tex] perfectly.
### 4. [tex]\( y = -\sqrt{x+5} - 3 \)[/tex]
Domain:
To ensure the expression under the square root is non-negative, we need:
[tex]\[ x + 5 \geq 0 \][/tex]
[tex]\[ x \geq -5 \][/tex]
Hence, the domain is [tex]\( x \geq -5 \)[/tex].
Range:
For [tex]\( x \geq -5 \)[/tex]:
[tex]\[ \sqrt{x + 5} \geq 0 \][/tex]
Thus,
[tex]\[ y = -\sqrt{x + 5} - 3 \leq 0 - 3 = -3 \][/tex]
So the range will be [tex]\( y \leq -3 \)[/tex].
This does not fit our required range of [tex]\( y \leq 3 \)[/tex].
### Conclusion:
The only function that satisfies both the domain [tex]\( x \geq 5 \)[/tex] and the range [tex]\( y \leq 3 \)[/tex] is:
[tex]\[ y = -\sqrt{x-5} + 3 \][/tex]
