\( \begin{array}{l}\text { Ile } 11.3 \text { More on Functions and Craphs } \\ \text { Part } 1 \text { of } 2\end{array} \quad \begin{array}{l}\text { Question } 17,3.59 \\ \begin{array}{l}\text { The domain of the piecewise function is }(-\infty, \infty) \\ \text { a. Graph the function. } \\ \text { b. Use your graph to determine the function's range. }\end{array}\end{array} \quad f(x)=\left\{\begin{array}{c}x \text { if } x<0 \\ -x \text { if } x \geq 0\end{array}\right] \)

Sure! Let's tackle **Question 17.3.59** step by step. ### **Given Function:** \[ f(x) = \begin{cases} x & \text{if } x < 0 \\ -\,x & \text{if } x \geq 0 \end{cases} \] ### **Part a: Graph the Function** To graph the piecewise function \( f(x) \), we'll consider each piece separately. 1. **For \( x < 0 \):** - **Function:** \( f(x) = x \) - **Graph:** This is a straight line with a slope of 1 passing through the origin. It extends to the left of the y-axis. 2. **For \( x \geq 0 \):** - **Function:** \( f(x) = -x \) - **Graph:** This is a straight line with a slope of -1, also passing through the origin. It extends to the right of the y-axis. **Visual Representation:** ``` | |\ | \ | \ ----------+---\---------- | \ | \ | \ | ``` - The **left side** (\( x < 0 \)) is the line with positive slope (\( f(x) = x \)). - The **right side** (\( x \geq 0 \)) is the line with negative slope (\( f(x) = -x \)). ### **Part b: Determine the Function's Range Using the Graph** The **range** of a function is the set of all possible output values (y-values). 1. **For \( x < 0 \):** - \( f(x) = x \) - As \( x \) approaches negative infinity, \( f(x) \) also approaches negative infinity. - The maximum y-value in this interval is just below 0 (since \( x < 0 \)). 2. **For \( x \geq 0 \):** - \( f(x) = -x \) - As \( x \) increases, \( f(x) \) decreases, approaching negative infinity. - The maximum y-value in this interval is 0 (when \( x = 0 \)). **Combining Both Segments:** - The highest point \( f(x) \) reaches is 0. - \( f(x) \) can take any value less than or equal to 0. **Therefore, the range of \( f(x) \) is:** \[ \boxed{\,(-\infty,\ 0]\,} \] ### **Summary:** - **Graph:** Consists of two straight lines, one with a positive slope for \( x < 0 \) and another with a negative slope for \( x \geq 0 \), both intersecting at the origin. - **Range:** All real numbers less than or equal to zero.