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Chapter 0: Problem 103

Explain the difference between a rational function and a polynomial function. Is every polynomial function a rational function? Why or why not?

Short Answer

Expert verified
A polynomial function is a special case of a rational function where the denominator is 1. Thus, every polynomial function is a rational function.

Step by step solution

01

Define a Polynomial Function

A polynomial function is a function of the form \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ \text{...} + a_1 x + a_0 \ \text{where } a_n, a_{n-1}, \text{...}, a_1, a_0 \ \text{are constants and } n \text{ is a non-negative integer.} \]
02

Define a Rational Function

A rational function is a function of the form \[ R(x) = \frac{P(x)}{Q(x)} \] where \(P(x)\) and \(Q(x)\) are polynomial functions and \(Q(x) eq 0\). This means that a rational function is essentially a fraction where both the numerator and the denominator are polynomials.
03

Comparison

The key difference is that a polynomial function is just a single polynomial expression, while a rational function is the ratio of two polynomial functions. Polynomial functions do not have a denominator, while rational functions do.
04

Is Every Polynomial Function a Rational Function?

Yes, every polynomial function \(P(x)\) can be considered a rational function. This is because it can be written as\[ \frac{P(x)}{1} \] where the denominator is the polynomial \(Q(x) = 1\), which is a valid polynomial.
05

Conclusion

Since the definition of a rational function includes the case where the denominator is the constant polynomial 1, every polynomial is inherently a rational function. Thus, every polynomial function is indeed a rational function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Function
Polynomial functions are essential building blocks in mathematics. They are expressions that consist of variables, coefficients, and exponents that are non-negative integers. A polynomial function typically looks like this: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 \] In this formula, each term's power decreases from left to right, starting from the highest power of \(x\) (which is \(n\)) down to the lowest.
Here are key features of polynomial functions:
  • All exponents are whole numbers.
  • Coefficients can be real or complex numbers.
  • There are no variables in the denominator.
  • They represent smooth, continuous curves when graphed.
Rational Function
Rational functions are a step beyond polynomial functions. They are fractions where both the numerator and the denominator are polynomials. 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 \(Q(x) eq 0\).
Important things to note about rational functions are:
  • The denominator polynomial must never be zero.
  • They can represent more varied shapes than polynomial functions, including asymptotes.
  • They allow for division of polynomials.
  • While polynomial functions are smooth, rational functions can have breaks or holes.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (like addition or multiplication). Polynomials and rational functions are types of algebraic expressions.
Here鈥檚 what you need to know about them:
  • Polynomials are algebraic expressions with multiple terms added together.
  • Rational functions are algebraic expressions written as a ratio of polynomials.
  • Variables can appear in various forms within these expressions but not under a radical sign for them to be considered polynomials or rational functions.
Mathematical Functions
Mathematical functions relate every input to a specific output. Both polynomial and rational functions are types of mathematical functions. Understanding their differences helps in solving algebraic problems more effectively.
Key points include:
  • A polynomial function is a type of mathematical function without a denominator.
  • A rational function, on the other hand, has both a numerator and a denominator that are polynomials.
  • Every polynomial function can be viewed as a special case of a rational function. This is because it can be written as \( \frac{P(x)}{1} \), where the denominator is simply 1.
  • Understanding how to manipulate and solve these functions is crucial in algebra.

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Most popular questions from this chapter

Red Tide is planning a new line of skis. For the first year, the fixed costs for setting up production are 45,000 dollars. The variable costs for producing each pair of skis are estimated at 80 dollars, and the selling price will be 255 dollars per pair. It is projected that 3000 pairs will sell the first year. a) Find and graph \(C(x),\) the total cost of producing \(x\) pairs of skis. b) Find and graph \(R(x),\) the total revenue from the sale of \(x\) pairs of skis. Use the same axes as in part (a). c) Using the same axes as in part (a), find and graph \(P(x),\) the total profit from the production and sale of \(x\) pairs of skis. d) What profit or loss will the company realize if the expected sale of 3000 pairs occurs? e) How many pairs must the company sell in order to break even?

Sally makes deposits into a retirement account every year from the age of 30 until she retires at age 65 . a) If Sally deposits 1200 per year and the account earns interest at a rate of \(8 \%\) per year, compounded annually, how much does she have in the account when she retires? (Hint: Use the annuity formula for Exercises 35 and 36 . ) b) How much of that total amount is from Sally's deposits? How much is interest?

The weight \(B\) of a human's brain is directly proportional to a person's body weight \(W.\) a) It is known that a person who weighs 120 lb has a brain that weighs 3 lb. Find an equation of variation expressing \(B\) as a function of \(W.\) b) Express the variation constant as a percent and interpret the resulting equation. c) What is the weight of the brain of a person who weighs 160 lb?

Suppose that \(\$ 3000\) is borrowed as a college loan, at \(5 \%\) interest, compounded daily, for \(t\) years. a) The amount \(A\) that is owed is a function of time. Find an equation for this function. b) Determine the domain of the function in part (a).

In computing the dosage for chemotherapy, a patient's body surface area is needed. A good approximation of a person's surface area \(s,\) in square meters \(\left(m^{2}\right),\) is given by the formula $$s=\sqrt{\frac{h w}{3600}},$$ where w is the patient's weight in kilograms (kg) and h is the patient's height in centimeters (cm). (Source: U.S. Oncology.) Use the preceding information. Round your answers to the nearest thousandth. Assume that a patient's weight is 70 kg. Approximate the patient's surface area assuming that: a) The patients height is 150 cm. b) The patients height is 180 cm.
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