Questions: Find the domain of the function. g(x) = sqrt(5x + 70) What is the domain of g? (Type your answer in interval notation.)

Find the domain of the function. g(x) = sqrt(5x + 70) What is the domain of g? (Type your answer in interval notation.)
Transcript text: Find the domain of the function. \[ g(x)=\sqrt{5 x+70} \] What is the domain of $g$ ? $\square$ (Type your answer in interval notation.)
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Solution

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Solution Steps

To find the domain of the function \( g(x) = \sqrt{5x + 70} \), we need to determine the values of \( x \) for which the expression under the square root is non-negative. This is because the square root function is only defined for non-negative numbers. Therefore, we solve the inequality \( 5x + 70 \geq 0 \).

Step 1: Understand the Function

The function given is \( g(x) = \sqrt{5x + 70} \). The square root function is defined only for non-negative values. Therefore, the expression inside the square root, \( 5x + 70 \), must be greater than or equal to zero.

Step 2: Set Up the Inequality

To find the domain, we need to solve the inequality:

\[ 5x + 70 \geq 0 \]

Step 3: Solve the Inequality

Subtract 70 from both sides:

\[ 5x \geq -70 \]

Next, divide both sides by 5:

\[ x \geq -14 \]

Step 4: Write the Domain in Interval Notation

The solution to the inequality \( x \geq -14 \) means that the domain of the function is all real numbers greater than or equal to \(-14\). In interval notation, this is written as:

\[ [-14, \infty) \]

Final Answer

The domain of \( g(x) \) is \(\boxed{[-14, \infty)}\).

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