Question: Part 3 of 5 0.66 of 10 points Points: 0.6 of 1 (a) Find the domain of the function. (b) Locate…

Part 3 of 5

0.66 of 10 points Points: 0.6 of 1

(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.

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. 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.

Locate any y-intercepts. Select the correct choice below and fill in any answer boxes within your choice.

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

Solution

(a) Find the domain of the function. The function \( f(x) \) is defined differently for \( x < 0 \) and \( x \geq 0 \). Thus, there is no restriction on \( x \). Therefore, the domain is all real numbers. (b) Locate any intercepts. X-intercepts: To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \). For \( x < 0 \), \( f(x) = 4 + x = 0 \). \[ 4 + x = 0 \] Subtract 4 from both sides: \[ x = -4 \] For \( x \geq 0 \), \( f(x) = x^2 \). \[ x^2 = 0 \] Take the square root of both sides: \[ x = 0 \] So, the x-intercepts are \( x = -4, 0 \). Y-intercepts: Set \( x = 0 \) in the function. For \( x = 0 \), \( f(x) = x^2 = 0^2 = 0 \). So, the y-intercept is \( y = 0 \). Final answers: - The domain is all real numbers. - The x-intercepts are \( x = -4 \) and \( x = 0 \). - There is one y-intercept: \( y = 0 \).