For each piecewise-defined function (a) calculate f(-1),...

For each piecewise-defined function
(a) calculate $f(-1), f(0), f(1),$ and $f(2),$ and (b) sketch a graph of $f$.
$f(x)=\left\{\begin{aligned} 3 x+1, & \text { if } x \leq 0 \\ 4, & \text { if } 0<x \leq 1 \\ x^{3}, & \text { if } x>1 \end{aligned}\right.$

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Key Concepts

Piecewise-Defined Functions

Piecewise-defined functions are functions that are described by different expressions over different parts of their domain. They require careful attention to the conditions specified for each subdomain to determine the correct formula to use when evaluating the function or sketching its graph.

Graphing Piecewise-Defined Functions

Graphing piecewise-defined functions entails plotting the distinct branches of the function according to the defined intervals. Key aspects include ensuring that the graph accurately reflects the correct formula on each interval, marking open or closed endpoints where the function may be discontinuous, and clearly indicating transitions between different pieces.

Function Evaluation

Function evaluation involves substituting specific input values into a function's expression to compute the corresponding output. In the context of a piecewise-defined function, it is essential to check which condition applies for the input value before performing the substitution.

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