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Chapter 2: Problem 3

What is the domain of the function \(f(x)=x^{2}-3 x ?\)

Short Answer

Expert verified
The domain of the function \(f(x) = x^{2} - 3x\) is all real numbers, or in interval notation, \((- \infty, + \infty)\).

Step by step solution

01

Identify Type of Function

Firstly, it is important to identify that the function \(f(x)=x^{2}-3x\) is a polynomial function. Polynomial functions are defined for all real numbers.
02

Determine the Domain

Since the function is a polynomial, the domain of this function is all real numbers. In interval notation, this is expressed as \((- \infty, + \infty)\), denoting that the function includes all values from negative infinity to positive infinity.

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Key Concepts

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

Polynomial Functions
A polynomial function is a type of mathematical expression consisting of variables and coefficients. It includes terms in the form of powers of the variable, generally written as follows:
  • Constant term: a number on its own.
  • Linear term: a variable raised to the power of one, such as \(x\).
  • Quadratic term: a variable squared, such as \(x^2\).
  • Cubic term: a variable cubed, such as \(x^3\).
Polynomials can have multiple terms, such as \(ax^n + bx^{n-1} + ... + c\), where \(a, b, c\) are constants. These terms are added together to form the polynomial.

In the exercise given, the function \(f(x) = x^2 - 3x\) includes a linear term \(-3x\) and a quadratic term \(x^2\). Because it is in polynomial form, it has no restrictions like divisions by zero or square roots of negative numbers, allowing it to be defined for all real numbers.
Real Numbers
Real numbers consist of a vast set of numbers that can be found on a number line. This includes both rational and irrational numbers, making the category very broad.
  • Rational numbers: numbers that can be expressed as a fraction, like \( \frac{1}{2} \) or 4.
  • Irrational numbers: numbers that cannot be expressed as a simple fraction, such as \(\sqrt{2}\) or \( \pi \).
Real numbers include all integer numbers, as well as decimals and fractions.

They are very crucial in defining domains for functions because they encompass all possible values the polynomial can take. When we say a polynomial function can be defined for all real numbers, this means that every point on the number line could plug into our function without causing any mathematical dilemmas.
Interval Notation
Interval notation is a mathematical method used to represent a range of numbers. It is especially useful in expressing the domain and range of functions in a compact form.

The notations use brackets and parentheses to indicate whether endpoints are included or not. Here's how the notation works:
  • Parentheses \((a,b)\): both endpoints \(a\) and \(b\) are not included.
  • Brackets \([a,b]\): both endpoints \(a\) and \(b\) are included.
  • Mixed \((a,b]\) or \([a,b)\): one endpoint is included while the other is not.

In the context of polynomial functions, which are defined for all real numbers, the domain is often written in interval notation as \((-\infty, +\infty)\).

This expression indicates that every real number between negative infinity and positive infinity is included, matching the fact that polynomial functions can take any real number as their input.

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Most popular questions from this chapter

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$g(t)=3 t^{2}-6 t-\frac{3}{4}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Without doing any calculations, explain why $$\sqrt{x+1}=-2$$ does not have a solution.

The following table gives the average hotel room rate for selected years from 1990 to \(1999 .\) (Source:American Hotel and Motel Association) $$\begin{array}{cc}\text { Year } & \text { Rate (in dollars) } \\\\\hline 1990 & 57.96 \\\1992 & 58.91 \\\1994 & 62.86 \\\1996 & 70.93 \\\1998 & 78.62 \\\1999 & 81.33\end{array}$$ (a) What general trend do you notice in these figures? (b) Fit both a linear and a quadratic function to this set of points, using the number of years since 1990 as the independent variable. (c) Based on your answer to part (b), which function would you use to model this set of data, and why? (d) Using the quadratic model, find the year in which the average hotel room rate will be \(\$ 85\)

Find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) for \(f(x)=a x+b,\) where \(a\) and \(b\) are constants.

A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available. (a) Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) Write an expression for the area of the plot in terms of its length. (d) Find the dimensions that will yield the maximum area.
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