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Polynomial Functions in One Variable Solutions

Polynomial Functions:

• Are in the form $p\left(x\right)=a_n x^n+a_{n–1} x^{n–1}+...+a_2 x^2+a_1 x+a_0$, in which:
  • $a_n \ne$   0   ,
  • n is a    whole    number, and
  • the coefficients $a_n, a_{n–1}, ..., a_2, a_1, a_0$are    real    numbers.
• With graphs that are    smooth and continuous   , draw without lifting the pencil or changing direction abruptly.
  • Smooth: no sharp points
  • Continuous: no breaks or gaps

Power Functions:

• Are in the form $w\left(x\right)=a{x}^n$, in which a and n are    real    numbers.
• Are monomial functions with a variable base raised to a    fixed    numerical power.
• May have a numerical    coefficient   .
• May have negative or fractional    exponents    (while a polynomial function cannot).

Example 1

Use the Venn diagram to sort the functions. Explain your reasoning.

$\begin{array}{cccc}y=x^2 & f\left(x\right)=x^{–\frac{1}{3}} & y=–4x^8+\sqrt{2} & h\left(x\right)=4\mathrm{\pi }{x}^3–8x+\mathrm{\pi }\\ g\left(x\right)=x{\left(1+x\right)}^{\frac{1}{2}}& y=\left|x–4\right|& y=\frac{1}{x}& k\left(x\right)=–\frac{1}{2}{x}^5\end{array}$

Example 2

Determine if the given functions represent a power and/or polynomial function, or neither.