Determine the domain of lower h left-parenthesis x right-parenthesis equals Start Fraction 5 over Start Root 2 x plus 10 End Root End Fraction and write it using set builder notation.

Question

(1 point)
Responses

left-brace x vertical-line x greater-than-or-equal-to negative 5 right-brace
Image with alt text: left-brace x vertical-line x greater-than-or-equal-to negative 5 right-brace

left-brace x vertical-line x greater-than-or-equal-to 5 right-brace
Image with alt text: left-brace x vertical-line x greater-than-or-equal-to 5 right-brace

left-brace x vertical-line x greater-than 5 right-brace
Image with alt text: left-brace x vertical-line x greater-than 5 right-brace

left-brace x vertical-line x greater-than negative 5 right-brace

Answer 1

To determine the domain of the function \( h(x) = \frac{5}{\sqrt{2x + 10}} \), we need to consider the constraints that arise from the expression inside the square root and the fact that the square root cannot be zero (as it appears in the denominator).

1. Inside the square root must be non-negative: \[ 2x + 10 > 0 \] Solving this inequality: \[ 2x > -10 \] \[ x > -5 \]

2. The square root must not be zero: \[ 2x + 10 \neq 0 \] Solving this equation: \[ 2x \neq -10 \] \[ x \neq -5 \]

Combining both results, we get that \( x \) must be strictly greater than -5.

Therefore, the domain of the function in set builder notation is: \[ { x \mid x > -5 } \]

Thus, the correct response is: left-brace x vertical-line x greater-than negative 5 right-brace.