Question: (a) Find the domain of the function. (b) Locate any intercepts. (c) Choose the correct graph of the…

(a) Find the domain of the function. (b) Locate any intercepts. (c) Choose the correct graph of the function. (d) Based on the graph, find the range.

(a) Find the domain of the function. Select the correct choice below and fill in any answer boxes within your choice.

A. The domain is {x | } (Type an inequality or a compound inequality.) B. The domain is all real numbers.

(b) Locate any x-intercepts. Select the correct choice below and fill in any answer boxes within your choice.

A. The x-intercept(s) is/are (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no x-intercepts.

Solution

To solve the problem, we need to analyze the function \( f(x) \) defined as: \[ f(x) = \begin{cases} 4 + x & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases} \] (a) Find the domain of the function. The function is defined for all \( x \) such that \( x < 0 \) or \( x \geq 0 \), which means the function is defined for all real numbers. Therefore, the domain is all real numbers. (b) Locate any x-intercepts. To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \). 1. For \( x < 0 \), \( f(x) = 4 + x = 0 \). \[ 4 + x = 0 \] Solve for \( x \): \[ x = -4 \] 2. For \( x \geq 0 \), \( f(x) = x^2 = 0 \). \[ x^2 = 0 \] Solve for \( x \): \[ x = 0 \] The x-intercepts are \( x = -4 \) and \( x = 0 \).