Question

Graph each piecewise-defined function.f(x)={6−xif x≤33if x\u003e3f(x) =
\begin{cases}
6 - x & 	ext{if } x \leq 3 \
3 & 	ext{if } x \u003e 3
\end{cases}

Accepted Answer

Step 1: Understand the piecewise function definition. The function is defined as $$f(x) = -x - 2$$ when $$x \leq 2$$, and $$f(x) = -4$$ when $$x \u003e 2. $$This means the graph will have two parts: one for $$x$$ values less than or equal to 2, and another for $$x$$ values greater than 2. Step 2: Graph the first piece $$f(x) = -x - 2$$ for $$x \leq 2. $$This is a linear function with slope $$-1$$ and y-intercept $$-2. $$Plot points for values of $$x$$ less than or equal to 2, for example at $$x=0$$, $$f(0) = -0 - 2 = -2$$, and at $$x=2$$, $$f(2) = -2 - 2 = -4. $$Connect these points with a straight line extending to the left. Step 3: Graph the second piece $$f(x) = -4$$ for $$x \u003e 2. $$This is a constant function, so for all $$x$$ values greater than 2, the function value is $$-4. $$Draw a horizontal line at $$y = -4$$ starting just to the right of $$x=2. $$Step 4: Determine the type of points at $$x=2. $$Since the first piece includes $$x=2 ($$because of $$\leq$$), plot a solid dot at $$(2, -4) on $$the line $$f(x) = -x - 2. $$For the second piece, since it is defined for $$x \u003e 2$$, do not include the point at $$x=2 on $$the constant line; instead, use an open circle at $$(2, -4) to $$indicate that this point is not included in the second piece. Step 5: Review the graph to ensure continuity and correct representation of the piecewise function. The graph should show a line with negative slope up to and including $$x=2$$, and a horizontal line at $$y=-4$$ for $$x$$ values greater than 2, with the appropriate solid and open points at $$x=2.$$

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} 6 - x & if x \leq 3 \\ 3 & if x > 3 \end{matrix}$

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Key Concepts

Piecewise-Defined Functions

Graphing Linear Functions

Domain Restrictions and Continuity

Graph each piecewise-defined function.f(x)={6−xif x≤33if x>3f(x) =\(\begin{cases}\)6 - x & \(\text{if }\) x \(\leq\) 3 \\3 & \(\text{if }\) x > 3\(\end{cases}\)