Question

Graph each piecewise-defined function. f(x)={−12x2+2if x≤212xif x\u003e2f(x) =
\begin{cases}
-\frac{1}{2}$$x^2 + 2$$ & 	ext{if } x \leq 2 \
\frac{1}{2}x & 	ext{if } x \u003e 2
\end{cases}

Accepted Answer

Identify the two pieces of the piecewise function and their respective domains: $$f(x) = x^2$ - 2$$ for $$x \leq 1$$ and $$f(x) = 2x$$ for $$x \u003e 1. $$Graph the first piece $$f(x) = x^2$ - 2$$ for all $$x$$ values less than or equal to 1. This is a parabola shifted down by 2 units. Plot points including $$x=1 to $$find the boundary value. Graph the second piece $$f(x) = 2x$$ for all $$x$$ values greater than 1. This is a straight line with slope 2. Start plotting from just greater than $$x=1 to $$the right. At $$x=1$$, evaluate both pieces to check if the function is continuous. Calculate $$f(1)$$ from the first piece and the limit from the second piece as $$x$$ approaches 1 from the right. Use open or closed circles to indicate whether the point at $$x=1 is $$included in the graph for each piece: closed circle for $$x \leq 1$$ and open circle for $$x \u003e 1. $$Then connect the points smoothly according to the function definitions.

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} - \frac{1}{2} x^{2} + 2 & if x \leq 2 \\ \frac{1}{2} x & if x > 2 \end{matrix}$

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Key Concepts

Piecewise-Defined Functions

Graphing Quadratic and Linear Functions

Domain Restrictions and Continuity

Graph each piecewise-defined function. f(x)={−12x2+2if x≤212xif x>2f(x) =\(\begin{cases}\)-\(\frac{1}{2}\)x^2 + 2 & \(\text{if }\) x \(\leq\) 2 \(\frac{1}{2}\)x & \(\text{if }\) x > 2\(\end{cases}\)