Define \(f(x)\) as follows
\[ f(x) = \begin{cases} 3x + 8 & \text { if } x \leq -1\\ -\frac {1}{3}(x-4) - 1 & \text { if } x > -1 \end{cases} \]

Graph of \(y = f(x)\):

Evaluate

\(f(-4) = \answer {-4}\)

\(f(-1) = \answer {5}\)

\(f(0) = \answer {\frac {1}{3}}\)

\(f(1) = \answer {0}\)

Domain and Range

The domain of \(f\) is

\((-\infty , 5)\) \((-\infty , 5]\) \((-\infty , -1) \cup (-1, \infty )\) \((-\infty , \infty )\)

The range of \(f\) is

\((-\infty , 1)\) \((-\infty , 1]\) \((-\infty , 5]\) \((-\infty , 5)\)
Zeros

List the zeros of \(f\), in numeric order.

\[ \answer {-\frac {8}{3}} \, \text { and } \, \answer {1} \]
Continuity

\(f\) is continuous, except at \(\answer {-1}\), which is a discontinuity singularity.

Behavior

\(f\) is increasing on \(\left ( \answer {-\infty }, \answer {-1} \right ]\).

\(f\) is decreasing on \(\left [ \answer {-1}, \answer {\infty } \right )\).

Extrema

The maximum value of \(f\) is \(\answer {5}\).

The minimum value of \(f\) is \(\answer {DNE}\).

2025-05-18 00:04:58