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Chapter 3: Problem 83

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Short Answer

Expert verified
No, every rational function is not a polynomial function because polynomial functions do not have variables in their denominator. However, every polynomial function is a rational function because every polynomial can be represented as a ratio where the denominator is 1.

Step by step solution

01

Defining Rational and Polynomial Functions

Rational functions are the ratio of two polynomial functions. The general form of a rational function is \(R(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\neq0\). Polynomial functions are a sum of terms involving powers of the same variable or variables. The general form of a polynomial function is \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2+ a_1x + a_0\), where \(a_n\neq0\).
02

Rational Functions as Polynomial Functions

In the general form of a rational function, a rational function can only be a polynomial function if the denominator \(q(x)\) is 1. If \(q(x)\) is anything else, then the rational function is not a polynomial function because polynomial functions do not have variables in the denominator.
03

Polynomial Functions as Rational Functions

In the general form of a rational function, if \(p(x)\) is a polynomial, and \(q(x)\) is 1, then every polynomial function is a rational function. This is because dividing any polynomial by 1 does not change the polynomial.

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Most popular questions from this chapter

The function \(f(x)=-0.00002 x^{3}+0.008 x^{2}-0.3 x\) \(+6.95\) models the number of annual physician visits, \(f(x),\) by a person of age \(x\) a. Graph the function for meaningful values of \(x\) and discuss what the graph reveals in terms of the variables described by the model. b. Use the zero or root feature of your graphing utility to find the age, to the nearest year, for the group that averages 13.43 annual physician visits. c. Verify part (b) using the graph of \(f\).

The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\) a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?

Can the graph of a polynomial function have no \(y\) -intercept? Explain,

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+4 x^{2}+4 x$$
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