To find the y-value for the function \( y = \sqrt{\frac{1}{2}x + 3} - 9 \) when \( x = 13 \), we will substitute \( x = 13 \) into the equation and simplify.
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Substitute \( x = 13 \):
\[ y = \sqrt{\frac{1}{2}(13) + 3} - 9 \]
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Calculate \( \frac{1}{2}(13) \):
\[ \frac{1}{2}(13) = 6.5 \]
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Add 3 to \( 6.5 \):
\[ 6.5 + 3 = 9.5 \]
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Now, take the square root of \( 9.5 \):
\[ \sqrt{9.5} \]
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Substitute back into the equation:
\[ y = \sqrt{9.5} - 9 \]
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Since \( \sqrt{9.5} \) is approximately \( 3.08 \):
\[ y \approx 3.08 - 9 \]
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Calculate the final value:
\[ y \approx 3.08 - 9 \approx -5.92 \]
Which is approximately \(-6\).
However, let's look at the choices given:
- −5
- 2
- −7
- −1
The closest value to \(-6\) is \(-5\), which is rounded up from about \(-5.92\).
Thus, the y-value when \( x = 13 \) is approximately −5.