In Exercises usefuncgraphfirst - usefuncgraphlast, use the graph of \(y = f(x)\) given below to answer the question.

[Picture]

Find the domain of \(f\). \(\answer {[-5,3]}\)
Find the range of \(f\). \(\answer {[-5,4]}\)
Find the maximum, if it exists.

\(f(-3) = 4\)
Find the minimum, if it exists.

\(f(-5) = -5\)
List the local maximums, if any exist.

\((-3,4)\), \((2,3)\)
List the local minimums, if any exist.

\((0,-1)\)
List the intervals where \(f\) is increasing.

\([-5,-3]\), \([0,2]\)
List the intervals where \(f\) is decreasing.

\([-3,0]\), \([2,3]\)
Determine \(f(-2)\). \(f(-2) = \answer {2}\)
Solve \(f(x) = 4\). \(x=\answer {-3}\)
List the \(x\)-intercepts, if any exist.

\((-4,0)\), \((-1,0)\), \((1,0)\)
List the \(y\)-intercepts, if any exist. \(\answer {(0,-1)}\)
Find the zeros of \(f\).

\(-4\), \(-1\), \(1\)
Solve \(f(x) \geq 0\).

\([-4,-1]\), \([1,3]\)
Find the number of solutions to \(f(x) = 1\). \(\answer {4}\)
Find the number of solutions to \(|f(x)| = 1\). \(\answer {6}\)
Solve \((x^2-x-2)f(x) = 0\)

\(x=-4, -1,1,2\)
Solve \((x^2-x-2)f(x) > 0\)

\((-4,-1) \cup (-1,1) \cup (2,3)\)
Find the domain of \(R(x) = \dfrac {1}{f(x)}\)

To find the domain of \(R(x) = \frac {1}{f(x)}\), we start with the domain of \(f\) and exclude values where \(f(x) = 0\). Hence, the domain of \(R\) is \([-5,-4) \cup (-4,-1) \cup (-1,1) \cup (1,3]\).
Find the range of \(R(x) = \dfrac {1}{f(x)}\)

To find the range of \(R(x) = \frac {1}{f(x)}\), we start with the range of \(f\) (excluding \(0\)) and take reciprocals. If \(-5 \leq y < 0\), then \(\frac {1}{y} \leq -\frac {1}{5}\). If \(0 < y \leq 4\), then \(\frac {1}{y} \geq \frac {1}{4}\). Hence the range of \(R\) is \(\left (-\infty , -\frac {1}{5} \right ] \cup \left [ \frac {1}{4}, \infty \right )\).
In Exercises usesecondfuncgraphfirst - usesecondfuncgraphlast, use the graph of \(y =g(t)\) given below to answer the question.

[Picture]

Find the domain of \(g\).

\([-4,4]\)
Find the range of \(g\).

\([-5,5)\)
Find the maximum, if it exists.

none
Find the minimum, if it exists.

\(g(-2) = -5\)
List the local maximums, if any exist.

none
List the local minimums, if any exist.

\((-2,-5)\), \((2,3)\)
List the intervals where \(g\) is increasing.

\([-2,2)\)
List the intervals where \(g\) is decreasing.

\([-4, -2]\), \((2,4]\)
Determine \(g(2)\).

\(g(2) = 3\)
Solve \(g(t) = -5\).

\(t=-2\)
List the \(t\)-intercepts, if any exist.

\((-4,0)\), \((0,0)\), \((4,0)\)
List the \(y\)-intercepts, if any exist.

\((0,0)\)
Find the zeros of \(g\).

\(-4\), \(0\), \(4\)
Solve \(g(t) \leq 0\).

\([-4,0] \cup \{4\}\)
Find the domain of \(G(t) = \dfrac {g(t)}{t+2}\).

\([-4,-2) \cup (-2.4]\)
Solve \(\dfrac {g(t)}{t+2} \leq 0\).

\(\{-4\} \cup (-2,0] \cup \{4\}\)
How many solutions are there to \([g(t)]^2 = 9\)?

\(5\)
Does \(g\) appear to be even, odd, or neither?

Neither.
Prove that if \(f\) is an odd function and \(0\) is in the domain of \(f\), then \(f(0) = 0\).
Let \(R(x)\) be the function defined as: \(R(x) = 1\) if \(x\) is a rational number, \(R(x) = 0\) if \(x\) is an irrational number. With help from your classmates, try to graph \(R\). What difficulties do you encounter?

NOTE: Between every pair of real numbers, there is both a rational and an irrational number …

Consider the graph of the function \(f\) given below.

[Picture]

Explain why \(f\) has a local maximum but not a local minimum at the point \((-1, 1)\).
Explain why \(f\) has a local minimum but not a local maximum at the point \((1, 1)\).
Explain why \(f\) has a local maximum AND a local minimum at the point \((0, 1)\).
Explain why \(f\) is constant on the interval \([-1, 1]\) and thus has both a local maximum AND a local minimum at every point \((x, f(x))\) where \(-1 < x < 1\).
For each function below, find the local maximum or local minimum and list the interval over which the function is increasing and the interval over which the function is decreasing.

  • [Picture]

    Local maximum: \((0,1)\), no local minimum. Increasing: \((0,2]\), decreasing: \([-2,0)\).

  • [Picture]

    No local maximum, local minimum: \((0,1)\). Increasing: \([-2,0)\), decreasing: \((0,2]\).

  • [Picture]

    No local maximum, local minimum: \((0,-1)\). Increasing: \([0,2]\), decreasing: \([-2,0]\).

  • [Picture]

    Local maximum: \((0,5)\), no local minimum. Increasing: \([-2,0]\), decreasing: \([0,2]\).