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Chapter 1: Problem 6

Find the domain of each function. $$f(x)=x^{2}+x-12$$

Short Answer

Expert verified
The domain of the function \(f(x)=x^{2}+x-12\) is all real numbers, which can be denoted as \((- \infty, \infty)\) in interval notation.

Step by step solution

01

Understand the Function Type

In this exercise, the function given is a polynomial function \(f(x)=x^{2}+x-12\). Polynomial functions are defined for all real numbers.
02

Determine the Domain

Since polynomial functions are defined for all real numbers, the domain of this function is all real numbers.
03

Write the Domain in Interval Notation

In interval notation, all real numbers can be represented as \((- \infty, \infty)\).

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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 mathematics. They are expressions made up of variables raised to whole number powers and combined with coefficients. An example of a polynomial function is the quadratic equation presented in the exercise: \[f(x) = x^2 + x - 12\]The highest power of the variable (in this case, 2) is known as the degree of the polynomial. Polynomial functions consist of terms like \(x^2\), \(x\), and constants like \(-12\). Key characteristics of polynomial functions include:
  • Smooth and continuous graphs.
  • The domain is usually all real numbers, meaning they can take any real value as an input.
  • Non-vertical asymptotes (except for linear polynomials).
Understanding how polynomial functions work is a crucial step in solving problems related to calculus and algebra.
Real Numbers
Real numbers are an essential building block in mathematics comprising all numbers that can be found on the number line. This includes:
  • Rational numbers, like fractions and whole numbers.
  • Irrational numbers, like \(\pi\) and \(\sqrt{2}\).
In the context of the exercise, real numbers are significant because the domain of a polynomial function like \(f(x) = x^2 + x - 12\) consists of all real numbers. Real numbers allow for the calculation and representation of all possible values the function \(f(x)\) can take. In essence, you can input any real number into the polynomial function, ensuring that it always produces a valid output.
Interval Notation
Interval notation is a method used to express a set of numbers along a number line. It is particularly useful when describing the domain of functions.For polynomial functions, which are defined for all real numbers, the domain is often expressed as \((-\infty, \infty)\) This simply means that the function accepts any value from negative infinity to positive infinity, covering all possible real numbers. When reading interval notation:
  • A parenthesis \(()\) indicates that the endpoint is not included in the interval. For example, \((-\infty, \infty)\) uses parentheses because infinity is not a concrete number that can be included.
  • Brackets \([]\) would signify that the endpoint is included. However, infinity is always expressed with parentheses.
This method of notation is compact and widely used in mathematics to describe domains, ranges, and sets of numbers comprehensively.

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Most popular questions from this chapter

In Exercises \(53-64,\) complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+12 x-6 y-4=0$$

Solve: \(\sqrt{x+10}-4=x\) (Section P.7, Example 12)

What must be done to a function's equation so that its graph is stretched vertically?

Perform the indicated operation or operations. $$(2 x-1)\left(x^{2}+x-2\right)$$

Does \((x-3)^{2}+(y-5)^{2}=0\) represent the equation of a circle? If not, describe the graph of this equation.
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