Question

Graph each piecewise-defined function. f(x)={x3+5if x≤0−x2if x\u003e0f(x) =
\begin{cases}
$$x^3 + 5$$ & 	ext{if } x \leq 0 \
$$-x^2$$ & 	ext{if } x \u003e 0
\end{cases}

Accepted Answer

First, carefully read the piecewise function and identify the domain for each piece. The function is given as:

$$f(x) = \begin{cases} x^3 + 5 & 	ext{if } x \leq 0 \ -x^2 & 	ext{if } x  0 (a $$common scenario), rewrite the function as:

$$f(x) = \begin{cases} x^3$ + 5 & 	ext{if } x \leq 0 \ -x^2$ & 	ext{if } x \u003e 0 \end{cases}$$

This will help in graphing each piece clearly. To graph the first piece $$f(x) = x^3$ + 5$$ for $$x \leq 0$$, create a table of values for several $$x$$ values less than or equal to zero, calculate corresponding $$f(x)$$ values, and plot these points. Remember that $$x^3 is a $$cubic function shifted up by 5 units. For the second piece $$f(x) = -x^2$ for x$$ \u003e 0$$, similarly create a table of values for positive x$ values, calculate f(x$$)$$, and plot these points. This is a downward-opening parabola starting from zero (since at x=0$, f(x) = 0$). Finally, combine both graphs, making sure to use a closed dot at x=0$ for the first piece (since it includes x=0$) and an open dot for the second piece at x=0$ (since it does not include x=0$).

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} x^{3} + 5 & if x \leq 0 \\ - x^{2} & if x > 0 \end{matrix}$

Verified video answer for a similar problem:

Key Concepts

Piecewise-Defined Functions

Domain and Inequality Notation

Graphing Polynomial Functions

Graph each piecewise-defined function. f(x)={x3+5if x≤0−x2if x>0f(x) =\(\begin{cases}\)x^3 + 5 & \(\text{if }\) x \(\leq\) 0 \\-x^2 & \(\text{if }\) x > 0\(\end{cases}\)