Questions: 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.)

Solution

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:

\[ [-14, \infty) \]