1. **State the problem:** We are given the graph of function $f(x)$, which forms a "V" shape centered at the origin $(0,0)$. We want to compare $f(x)$ to the function $$g(x) = -2|x - 3|$$ and determine which statement about their intercepts, domain, or range is correct. 2. **Analyze $f(x)$ from the graph:** - The vertex of $f(x)$ is at the origin $(0,0)$. - Since the graph forms a "V" shape opening upwards and downwards, let's identify the exact equation shape. - The left line passes through $(-2,4)$ and $(-4,8)$, so slope is $$m = \frac{8 - 4}{-4 + 2} = \frac{4}{-2} = -2.$$ The right line passes through $(2,4)$ and $(4,8)$ with slope $$m = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2.$$ - This means: - For $x < 0$, $f(x) = -2x$ (lines with negative slope) - For $x > 0$, $f(x) = 2x$ (line with positive slope) - The function $f(x)$ can be described as $$f(x) = 2|x|.$$ Its vertex is at $(0,0)$. 3. **Analyze $g(x) = -2|x - 3|$:** - The vertex is at $x=3$ because of $|x-3|$. - $g(3) = -2|3-3| = 0$ so the vertex is at $(3,0)$. - The graph opens downward because of the negative coefficient $-2$. 4. **Compare the statements:** - **Same x-intercept?** - $f(x)$ has x-intercept where $f(x) = 0$, which occurs at $x=0$. - $g(x)$ has x-intercept where $g(x)=0$, at $x=3$. - No, different x-intercepts. - **Same y-intercept?** - $f(0) = 2|0| = 0$, so y-intercept is $0$. - $g(0) = -2|0-3| = -2| -3| = -2\times 3 = -6$, y-intercept is $-6$. - No, different y-intercepts. - **Same domain?** - Both are absolute value functions, defined for all real numbers. - Yes, both have domain $(-\infty, \infty)$. - **Same range?** - $f(x) = 2|x|$ yields values $\geq 0$ (range $[0, \infty)$). - $g(x) = -2|x-3|$ yields values $\leq 0$ (range $(-\infty, 0]$). - No, different ranges. 5. **Conclusion:** The correct statement is **The function $f(x)$ has the same domain as $g(x)$**. Final answer: The function $f(x)$ and the function $g(x) = -2|x - 3|$ have the same domain.