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Which function has a domain of [tex]x \geq 5[/tex] and a range of [tex]y \leq 3[/tex]?

A. [tex]y = \sqrt{x - 5} + 3[/tex]
B. [tex]y = \sqrt{x + 5} - 3[/tex]
C. [tex]y = -\sqrt{x - 5} + 3[/tex]
D. [tex]y = -\sqrt{x + 5} - 3[/tex]

Sagot :

GinnyAnswer
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]
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