Question

the polynomial function ( f ) is an odd function with domain ( -4leq xleq4 ). the table gives information about values of ( f(x) ) and the behavior of the function.
what is the absolute maximum value of ( f ) on its domain?

Explanation:

Step1: Recall the property of odd function

An odd function satisfies \(f(-x)=-f(x)\). The domain is \(-4\leq x\leq4\). We can use the symmetry of the odd - function.

Step2: Analyze the given values

We know the values for \(0 < x\leq4\). For an odd function, if \(x\) is in the domain, then \(-x\) is also in the domain.
When \(x = 4\), \(f(4)=2\). When \(x=- 4\), by the property of odd function \(f(-4)=-f(4)=- 2\).
When \(x = 3\), \(f(3)=0\). When \(x=-3\), \(f(-3)=-f(3)=0\).
When \(x = 2\), \(f(2)=-3\). When \(x=-2\), \(f(-2)=-f(2)=3\).
When \(x = 1\), \(f(1)=-1\). When \(x=-1\), \(f(-1)=-f(1)=1\).

Step3: Determine the absolute maximum

The absolute maximum value of a function \(y = f(x)\) on an interval \([a,b]\) is the largest value of \(y\) in that interval.
We check all the values: \(f(-4)=-2\), \(f(-3)=0\), \(f(-2)=3\), \(f(-1)=1\), \(f(0)=0\), \(f(1)=-1\), \(f(2)=-3\), \(f(3)=0\), \(f(4)=2\)

Answer:

\(3\) (corresponding to option B)