Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 44
Identify each function as a polynomial, a rational function, an exponential function, a piecewise linear function, or none of these. (Do not graph them; just identify their types.) $$ f(x)=x^{3}-x^{2 / 3} $$
Short Answer
Expert verified
The function is none of these categories.
Step by step solution
01
Define Polynomial Function
A polynomial function is a mathematical expression consisting of variables (also known as indeterminates) raised to non-negative integer exponents and multiplied by coefficients. An example is \( f(x) = x^3 - 2x^2 + 4 \).
02
Define Rational Function
A rational function is a ratio of two polynomials, where the denominator polynomial is not zero. For example, \( f(x) = \frac{x^2 + 1}{x - 3} \) is a rational function.
03
Define Exponential Function
An exponential function is of the form \( f(x) = a \, b^x \), where the base \( b \) is a constant and \( x \) appears in the exponent. An example is \( f(x) = 2^x \).
04
Define Piecewise Linear Function
A piecewise linear function is defined by different linear expressions for different intervals of the variable's domain. It looks like multiple lines connected together over these intervals.
05
Define None of These
If a function cannot be classified under the categories of polynomial, rational, exponential, or piecewise linear, it is labeled as 'none of these'.
06
Analyze Given Function
The function given is \( f(x) = x^3 - x^{2/3} \). To determine its type, observe the terms. The first term, \( x^3 \), is a polynomial term (non-negative integer exponent), but the second term, \( x^{2/3} \), has a fractional exponent, which disqualifies \( f(x) \) from being a polynomial.
07
Determine Function Type
Since \( f(x) = x^3 - x^{2/3} \) has a non-integer exponent and cannot be expressed as a ratio of polynomials, an exponential function, or a piecewise linear function, it does not fit any of the defined categories.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Function
Polynomial functions are very common in mathematics and are quite straightforward to identify. They are sums of terms, each consisting of a coefficient multiplied by a variable raised to a non-negative integer power. This means expressions like \( x^5 - 4x^2 + x - 7 \) are polynomial functions. Such functions are smooth, continuous, and can be drawn without lifting a pencil from the paper. Key features include:
- All exponents must be whole numbers (integers greater than or equal to zero).
- The coefficients can be any real number.
- There should be no variables in the denominator, nor should there be fractional exponents or negative exponents.
Rational Function
A rational function is like a fraction in mathematics, consisting of a numerator and a denominator, both of which are polynomial functions. The key characteristic of rational functions is that the denominator cannot be zero, as division by zero is undefined in mathematics. For example, \( f(x) = \frac{x^2 + 4}{x - 2} \) is a rational function as long as \( x eq 2 \). Important points include:
- The numerator and denominator must be polynomials.
- Rational functions can illustrate asymptotic behavior, meaning they can have vertical and horizontal asymptotes.
Exponential Function
Exponential functions hold a special place in mathematics due to their unique growth properties. They can be identified by the presence of a constant base raised to a variable exponent. Basic forms look like \( f(x) = a \cdot b^x \), where \( b \) is a positive constant and \( x \) is the variable in the exponent. Here are some characteristics:
- The base \( b \) should not be equal to 1, as this returns a constant function.
- Exponential functions grow rapidly and can model a wide range of real-world phenomena, like population growth or radioactive decay.
Piecewise Linear Function
Piecewise linear functions are like mathematical patchwork quilts. They are composed of different linear segments stitched together across various intervals of the domain. This design allows piecewise linear functions to model situations where a single linear equation doesn't fit all conditions. Key characteristics include:
- Defined by different linear subsidiaries for different input ranges.
- Commonly used to represent situations with distinct phases or conditions, like different tax brackets or distance vs. time graphs in physics.
