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Chapter 0: Problem 36

Identify the given function as polynomial, rational, both or neither. $$f(x)=3-2 x+x^{4}$$

Short Answer

Expert verified
The given function \(f(x)=3-2x+x^4\) is a polynomial function

Step by step solution

01

Determine whether the function is a Polynomial

Check if the function involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In this case, \(f(x)=3-2x+x^4\) follows all these criteria: it has addition and subtraction, any multiplication is between variables and coefficients, and the power of the variable x (i.e., its exponent) is a non-negative integer (in this case, 4 and 1). Therefore, \(f(x)=3-2x+x^4\) is a polynomial function.
02

Determine whether the function is a Rational function

A rational function is any function which can be defined as the quotient of two polynomial functions. Since \(f(x)=3-2x+x^4\) is not expressed as a ratio of two polynomial expressions, it can't be classified as a rational function.

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Key Concepts

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

Rational Functions
Rational functions are an important category in mathematics. They are functions that can be written as the division of two polynomials. This means you will often see them in the form \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are both polynomials, and \( Q(x) \) is not equal to zero. When analyzing a function to determine if it is rational, check for these key features:
  • It is composed of two polynomial expressions.
  • The function is written as a fraction.
For example, \( \frac{x^2 + 1}{x - 3} \) is a rational function. Watch out for zeros in the denominator because they affect the domain of the function. Rational functions often have asymptotes and discontinuities.
Function Classification
Let's delve into the broader idea of function classification. Classifying functions is about recognizing the general form based on their algebraic expressions;
  • If a function involves addition or subtraction, it's likely a polynomial.
  • If there is division between two polynomials, it is rational.
  • Functions involving roots or trigonometric terms fall into different categories.
The process of classification helps in predicting the features and the behavior of the function. In our exercise, the classification simplified it to a polynomial. Recognizing polynomial from other types like exponential or logarithmic can help avoid potential pitfalls in solving equations or graphing.
Algebraic Expressions
Algebraic expressions form the building blocks for functions in mathematics. They are combinations of variables, numbers, and operations (addition, subtraction, multiplication, etc.).
Through polynomials, one of the most fundamental algebraic expressions, we utilize integers to create terms like \( 3x \) or \( x^4 \).
  • Variables in algebraic expressions represent numbers and can change values.
  • Numbers are constants providing fixed values in expressions.
  • Operations between them define the structure of these expressions.
Learning to manipulate and understand algebraic expressions is key, as these skills apply to understanding and solving equations, systems of equations, and beyond.

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Most popular questions from this chapter

Use a graphing calculator to graph \(y=x e^{-x}, y=x e^{-2 x}\) \(y=x e^{-3 x}\) and so on. Estimate the location of the maximum for each. In general, state a rule for the location of the maximum of \(y=x e^{-k x}\)

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{4}-2 x+1$$

A picture hanging in an art gallery has a frame 20 inches high, and the bottom of the frame is 6 feet above the floor. A person whose eyes are 6 feet above the floor stands \(x\) feet from the wall. Let \(A\) be the angle formed by the ray from the person's eye to the bottom of the frame and the ray from the person's eye to the top of the frame. Write \(A\) as a function of \(x\) and graph \(y=A(x)\) GRAPH CANT COPY

Discuss whether the function described has an inverse. Suppose that your boss informs you that you have been awarded a \(10 \%\) raise. The next week, your boss announces that due to circumstances beyond her control, all employees will have their salaries cut by \(10 \% .\) Are you as well off now as you were two weeks ago? Show that increasing by \(10 \%\) and decreasing by \(10 \%\) are not inverse processes. Find the inverse for adding \(10 \%\) (Hint: To add \(10 \%\) to a quantity you can multiply the quantity by \(1.10 .)\)

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{3}-3 x+1$$
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