Question

12 part of a cubic function is graphed. identify the domain and range of only the part shown using interval notation
domain
range
how would your answer be different if both circles were open instead of closed?
domain
range

1. define the following vocabulary terms. use your book!!

polynomial function
symmetric about a line
symmetric about a point
power function
even function
odd function
absolute minimum
absolute maximum
extrema

Explanation:

• Polynomial function: A function of the form \(f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0}\), where \(n\) is a non - negative integer and \(a_{i}\) are real numbers.
• Symmetric about a LINE: A graph is symmetric about a line \(L\) if for every point \(P(x,y)\) on the graph, the reflection of \(P\) across the line \(L\) is also on the graph. For example, the graph of \(y = x^{2}\) is symmetric about the \(y\) - axis (\(x = 0\)).
• Symmetric about a POINT: A graph is symmetric about a point \(C(h,k)\) if for every point \(P(x,y)\) on the graph, the point \(Q(2h - x,2k - y)\) (the point symmetric to \(P\) with respect to \(C\)) is also on the graph. The graph of \(y=x^{3}\) is symmetric about the origin \((0,0)\).
• Power function: A function of the form \(y = ax^{b}\), where \(a\) and \(b\) are real numbers. For example, \(y = 3x^{2}\), \(y=\frac{1}{2}x^{-1}\) are power functions.
• Even function: A function \(f(x)\) such that \(f(-x)=f(x)\) for all \(x\) in the domain of \(f\). The graph of an even function is symmetric about the \(y\) - axis. For example, \(y = x^{2}\), \(y=\cos x\) are even functions.
• Odd function: A function \(f(x)\) such that \(f(-x)=-f(x)\) for all \(x\) in the domain of \(f\). The graph of an odd function is symmetric about the origin. For example, \(y = x^{3}\), \(y=\sin x\) are odd functions.
• Absolute Minimum: The smallest value of the function \(y = f(x)\) over its entire domain (or a specified interval). If \(f(c)\leq f(x)\) for all \(x\) in the domain (or interval) \(D\), then \(f(c)\) is the absolute minimum and \(c\) is the point of absolute minimum.
• Absolute Maximum: The largest value of the function \(y = f(x)\) over its entire domain (or a specified interval). If \(f(c)\geq f(x)\) for all \(x\) in the domain (or interval) \(D\), then \(f(c)\) is the absolute maximum and \(c\) is the point of absolute maximum.
• Extrema: The absolute maximum and absolute minimum values of a function (collectively). They can also refer to local maximum and local minimum values (in the context of calculus, when considering intervals).

Answer:

• Polynomial function: \(f(x)=a_{n}x^{n}+a_{n - 1}x^{n-1}+\cdots+a_{1}x + a_{0},n\in\mathbb{Z}_{\geq0},a_{i}\in\mathbb{R}\)
• Symmetric about a LINE: For every point \(P(x,y)\) on the graph, its reflection across the line is also on the graph
• Symmetric about a POINT: For every point \(P(x,y)\) on the graph, \(Q(2h - x,2k - y)\) (w.r. to point \((h,k)\)) is on the graph
• Power function: \(y = ax^{b},a,b\in\mathbb{R}\)
• Even function: \(f(-x)=f(x)\), symmetric about \(y\) - axis
• Odd function: \(f(-x)=-f(x)\), symmetric about origin
• Absolute Minimum: Smallest function value over domain (or interval)
• Absolute Maximum: Largest function value over domain (or interval)
• Extrema: Absolute (or local) maxima and minima