Question

Evaluate \(f\left( { - {\bf{3}}} \right)\), \(f\left( {\bf{0}} \right)\), and \(f\left( {\bf{2}} \right)\) for the piecewise defined function. Then sketch the graph of the function.

\(f\left( x \right) = \left\{ {\begin{aligned}{x + {\bf{1}}}&{{\bf{if}}\;\;x \le - {\bf{1}}}\\{{x^2}}&{{\bf{if}}\;\;x > - {\bf{1}}}\end{aligned}} \right.\)

Short answer

\(f\left( { - 3} \right) = - 2\), \(f\left( 0 \right) = 0\), and \(f\left( 2 \right) = 4\)

Step-by-step solution

Find the value of the function for \(x \le  - {\bf{1}}\)

For \(x \le 1\), the function is \(f\left( x \right) = x + 1\).

The value of \(f\left( { - 3} \right)\)is calculated below:

\(\begin{aligned}f\left( { - 3} \right) &= - 3 + 1\\ &= - 2\end{aligned}\)

Find the value of the function for \(x >  - {\bf{1}}\)

For \(x > - 1\), the function is \(f\left( x \right) = {x^2}\).

The value of \(f\left( 0 \right)\)is calculated below:

\(\begin{aligned}f\left( 0 \right) &= {\left( 0 \right)^2}\\ &= 0\end{aligned}\)

The value of \(f\left( 2 \right)\)is calculated below:

\(\begin{aligned}f\left( 2 \right) &= {\left( 2 \right)^2}\\ &= 4\end{aligned}\)

Sketch the graph of \(f\left( x \right)\)

The graph of \(f\left( x \right)\) is shown below.