1. **State the problem:** We have a piecewise function defined as $$f(x) = \begin{cases} -2x & \text{if } x < 0 \\ 2x & \text{if } x \geq 0 \end{cases}$$ We need to graph this function and determine its range from the graph. 2. **Analyze each piece:** - For $x < 0$, $f(x) = -2x$. Since $x$ is negative here, $-2x$ becomes positive (because negative times negative is positive). This part is a line with positive slope $2$ but only for negative $x$ values. - For $x \geq 0$, $f(x) = 2x$, which is a line with positive slope $2$ for non-negative $x$. 3. **Check continuity at $x=0$:** - From the left, $f(0^-) = -2 \times 0 = 0$ - From the right, $f(0) = 2 \times 0 = 0$ So the function is continuous at $x=0$. 4. **Graph shape:** - For $x<0$, the line $y = -2x$ is positive and increasing as $x$ moves left to zero. - For $x \geq 0$, the line $y = 2x$ is also increasing from zero upwards. - Together, these form a "V" shape opening upwards with vertex at the origin. 5. **Range determination:** - Since both parts produce non-negative values and the minimum value at $x=0$ is $0$, the range is all real numbers $y$ such that $$y \geq 0$$ 6. **Answer to part a:** The correct graph is the one showing a "V" shape opening upward with vertex at the origin, which corresponds to option C. 7. **Answer to part b:** The range of the function is $$[0, \infty)$$