Sketch the graph of the given function, evaluate the given...

Sketch the graph of the given function, evaluate the given expressions, and then use technology to duplicate the graphs. Give the technology formula. Sketch the graph of the given function, evaluate the given expressions, and then use technology to duplicate the graphs. Give the technology formula. $f(x)=\left\{\begin{array}{ll}-1 & \text { if }-4 \leq x \leq 0 \\ x & \text { if } 0<x \leq 4\end{array}\right.$ a. $f(-1)$ b. $f(0)$ c. $f(1)$

Sketch the graph of the given function, evaluate the given expressions, and then use technology to duplicate the graphs. Give the technology formula.
Sketch the graph of the given function, evaluate the given expressions, and then use technology to duplicate the graphs. Give the technology formula.
$f(x)=\left\{\begin{array}{ll}-1 & \text { if }-4 \leq x \leq 0 \\ x & \text { if } 0<x \leq 4\end{array}\right.$
a. $f(-1)$
b. $f(0)$
c. $f(1)$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different intervals in its domain. It assigns different formulas to different parts of its domain, and understanding the conditions under which each formula applies is crucial to correctly evaluating and graphing the function.

Domain and Interval Notation

The domain of a function specifies the set of input values for which the function is defined. When working with piecewise functions, each piece has its own subdomain typically defined using interval notation, ensuring clarity on which formula to use for a given input.

Function Evaluation

Evaluating a function involves substituting a given input into the appropriate formula based on the input's placement within the function's subdomains. For piecewise functions, it is essential to identify the correct interval to determine which expression to use when substituting a specific value.

Graphing Piecewise Functions

Graphing a piecewise function requires plotting each segment of the function within its defined interval. It involves paying attention to the boundaries of each interval, using open or closed endpoints as dictated by the inclusion or exclusion of endpoints in the interval definitions.

Technology Integration in Graphing

Using technology to graph functions, including piecewise functions, involves inputting the function definitions into graphing software or calculators. Typically, this involves using conditional statements or built-in piecewise function syntax to accurately replicate the function's segments, which aids in verifying manual sketches and calculations.

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