To bring this back to our original purpose, let's restate our definition of a rational function. [br][br]A [b]rational function[/b] [math]r[/math] is a function that can be expressed in the form [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math], where [math]p[/math] is a polynomial and [math]q[/math] is a non-zero polynomial.[br][br]A [b]polynomial function[/b] [math]f[/math] is a function of the form [math]f\left(x\right)=c_nx^n+c_{n-1}x^{n-1}+...+c_1x+c_0[/math] where each of the [math]c_i[/math]s are real numbers and [math]n[/math] is a non-negative integer.[br][br]
Which of the following functions are polynomials?
Which of the following functions are rational?
State a general relationship between polynomial functions and rational functions. (Hint: think squares and rectangles.)
All polynomial functions are rational functions, but not all rational functions are polynomial functions.
A key distinction between polynomial and rational functions is that, while all polynomials are continuous, not all rational functions are continuous. Plot a rational function below that has a discontinuity at [math]x=1[/math].
If [math]r\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}[/math] is a rational function with a discontinuity at [math]x=3[/math], what can you conclude?
[math]q\left(3\right)=0[/math]; i.e., [math]3[/math] is a zero of the polynomial [math]q[/math]