(a) Find the domain of each function. (b) Locate any interepts. (c) Graph each funct

step 1 We have been given a piecewise function F of X. We have been given a piecewise function F of X,

(a) Find the domain of each function. (b) Locate...

$$ \begin{array}{ll}{\text { (a) Find the domain of each function. }} & {\text { (b) Locate any interepts. }}& {\text {(c) Graph each function.. }} \\ {\text { (d) Based on the graph, find the range. }} & {\text { (e) Is feontinuous on its domain? }}\end{array} $$ $$ f(x)=\left\{\begin{array}{ll}{|x|} & {\text { if }-2 \leq x<0} \\ {x^{3}} & {\text { if } x>0}\end{array}\right. $$

$$
\begin{array}{ll}{\text { (a) Find the domain of each function. }} & {\text { (b) Locate any interepts. }}& {\text {(c) Graph each function.. }} \\ {\text { (d) Based on the graph, find the range. }} & {\text { (e) Is feontinuous on its domain? }}\end{array}
$$
$$
f(x)=\left\{\begin{array}{ll}{|x|} & {\text { if }-2 \leq x<0} \\ {x^{3}} & {\text { if } x>0}\end{array}\right.
$$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. Understanding piecewise functions involves recognizing how the definition changes at the boundaries between pieces and ensuring that the function’s behavior is clearly analyzed within each segment of the domain.

Domain of Functions

The domain of a function is the set of all possible inputs for which the function is defined. Determining the domain involves considering restrictions such as division by zero, taking even roots of negative numbers, or any other restrictions imposed by the function’s expressions.

Range of Functions

The range of a function is the set of all possible output values it can produce. Finding the range involves analyzing the behavior of the function across its entire domain, which may require studying different segments separately if the function is piecewise.

Intercepts

Intercepts are the points where the function crosses the axes: the x-intercepts are the points where the function output is zero, and the y-intercept is where the input is zero. Identifying these intercepts often simplifies the process of graphing the function and aids in understanding its overall behavior.

Graphing Functions

Graphing a function involves plotting its inputs and corresponding outputs to visually represent its behavior. In the context of piecewise functions, it is important to graph each piece accurately according to its defined interval, and to pay careful attention to endpoints and open or closed dots to indicate continuity or discontinuity at those boundaries.

Continuity

Continuity of a function refers to the property that small changes in the input result in small changes in the output, without any abrupt jumps or holes. When examining continuity, especially for piecewise functions, it is crucial to check the transition points between different function pieces to determine if the function is continuous at those boundaries.

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