Questions: Which function has a domain of (x geq 5) and a range of (y leq 3) ? - (y=sqrtx-5+3) - (y=sqrtx+5-3) - (y=-sqrtx-5+3) - (y=-sqrtx+5-3)

Solution Steps

Given the functions: \[ \begin{array}{l} y = \sqrt{x-5} + 3 \\ y = \sqrt{x+5} - 3 \\ y = -\sqrt{x-5} + 3 \\ y = -\sqrt{x+5} - 3 \end{array} \]

We need to determine which function has a domain of \( x \geq 5 \) and a range of \( y \leq 3 \).

1. For \( y = \sqrt{x-5} + 3 \):

   • The domain is \( x \geq 5 \) because the argument of the square root must be non-negative.

2. For \( y = \sqrt{x+5} - 3 \):

   • The domain is \( x \geq -5 \) because the argument of the square root must be non-negative.

3. For \( y = -\sqrt{x-5} + 3 \):

   • The domain is \( x \geq 5 \) because the argument of the square root must be non-negative.

4. For \( y = -\sqrt{x+5} - 3 \):

   • The domain is \( x \geq -5 \) because the argument of the square root must be non-negative.

1. For \( y = \sqrt{x-5} + 3 \):

   • The range is \( y \geq 3 \) because the square root function is always non-negative and adding 3 shifts it upwards.

2. For \( y = \sqrt{x+5} - 3 \):

   • The range is \( y \geq -3 \) because the square root function is always non-negative and subtracting 3 shifts it downwards.

3. For \( y = -\sqrt{x-5} + 3 \):

   • The range is \( y \leq 3 \) because the square root function is always non-negative, negating it makes it non-positive, and adding 3 shifts it upwards.

4. For \( y = -\sqrt{x+5} - 3 \):

   • The range is \( y \leq -3 \) because the square root function is always non-negative, negating it makes it non-positive, and subtracting 3 shifts it downwards.

We need a function with:

• Domain: \( x \geq 5 \)
• Range: \( y \leq 3 \)

From the analysis:

• \( y = -\sqrt{x-5} + 3 \) has the correct domain \( x \geq 5 \) and the correct range \( y \leq 3 \).

Final Answer