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Chapter 3: Problem 1

Polynomial and rational functions are examples of _____________ functions.

Short Answer

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Polynomial and rational functions are examples of Algebraic Functions.

Step by step solution

01

Understanding Polynomial and Rational Functions

A Polynomial function is characterized by an equation consisting of variables and coefficients. It involves sums of powers of variables multiplied by coefficients, for example: \(f(x) = 2x^3 + 3x^2 - 4x + 1\). \n\nRational functions, on the other hand, are fractions of polynomials. In other words, they have a polynomial in the numerator and/or denominator. For example: \(f(x) = (x^2 + 1) / (x - 3)\). Therefore, what Polynomial and Rational functions have in common is being a mathematical function.
02

Identifying the Category

From the explained definitions, it's clear that both polynomial and rational functions are examples of algebraic functions. In general, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite simply, it is a function that involves only algebraic operations like addition, subtraction, multiplication, and division, as well as application of the functions of the first or a higher degree. Therefore both polynomial and rational functions fall under this category.

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Key Concepts

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

Polynomial Functions
Polynomial functions are a fundamental concept in algebra. They are formed by summing multiples of powers of variables. In simpler terms, they consist of terms with a variable raised to a whole number exponent, each multiplied by a coefficient. For instance, a polynomial function can look like this: \( f(x) = 4x^5 + 2x^4 - 3x + 7 \). Here,
  • \(4x^5\), \(2x^4\), and \(-3x\) are individual terms.
  • The numbers 4, 2, -3, and 7 are coefficients.
  • \(x\) is the variable.
  • The powers of \(x\) are called exponents, such as 5, 4, and 1 in this example.
Polynomial functions can be as simple as a constant, like \( f(x) = 7 \), or more complex like the example above.
They are widely used because they provide simple approximations of complex problems and appear in many real-world models.
Rational Functions
Rational functions extend the versatility of polynomial functions by creating fractions from them. The defining trait is that they are expressions that represent the ratio of two polynomial functions. For example, a rational function may appear as: \( f(x) = \frac{2x^3 + x - 1}{x^2 - 4} \). In this function:
  • "2x^3 + x - 1" is the numerator, which is a polynomial.
  • "x^2 - 4" is the denominator, another polynomial.
Rational functions can generate a wide range of behaviors, such as asymptotes and holes in their graphs, due to the division involved in their structure.
They are applicable in situations where relationships are naturally expressed as ratios, such as rates of change, and they model complex scenarios with greater flexibility than simple polynomial functions.
Function Operations
Function operations allow us to create new functions by combining existing ones using basic arithmetic. These include:
  • Addition: \((f + g)(x) = f(x) + g(x)\)
  • Subtraction: \((f - g)(x) = f(x) - g(x)\)
  • Multiplication: \((f \cdot g)(x) = f(x) \cdot g(x)\)
  • Division: \((\frac{f}{g})(x) = \frac{f(x)}{g(x)}, \text{provided } g(x) eq 0\)
Function operations are foundational techniques that allow us to manipulate and understand complex mathematical expressions. They provide the tools necessary for evaluating how two different functions might interact or relate to one another.
These operations are essential for solving equations and modeling more intricate systems with multiple variables.

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Most popular questions from this chapter

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$e^{0.09 t}=3$$

Find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline y & 1 & 1.587 & 2.080 & 2.520 & 2.924 & 3.302 \\ \hline\end{array}$$

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{2} x$$

Condense the expression to the logarithm of a single quantity. $$\frac{1}{2}\left[\log _{4}(x+1)+2 \log _{4}(x-1)\right]+6 \log _{4} x$$

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$e^{-2 x}-2 x e^{-2 x}=0$$
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