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Chapter 1: Problem 5

Find the domain of each function. $$f(x)=x^{2}-2 x-15$$

Short Answer

Expert verified
The domain of the function \(f(x) = x^{2}-2x-15\) is all real numbers or \((-鈭�, 鈭�)\).

Step by step solution

01

Identify the function type

Firstly, observe the given function. It is \( f(x) = x^{2}-2x-15 \), a quadratic function. Because there are no restrictions like square roots or divisions by a variable, x can be any real number.
02

Determine the domain

Since x has no restrictions and can take on any value, the domain of this function is all real numbers. In interval notation, that is often written as \((-鈭�, 鈭�)\).

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Key Concepts

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

Domain of a Function
The domain of a function is simply all the possible values that the independent variable (usually represented as \( x \)) can take. Essentially, it tells us "where" the function is defined. For many common functions, like quadratic functions, the domain is unrestricted.
For instance, a quadratic function like \( f(x) = x^{2}-2x-15 \) can accept any real number as an input. This means, there are no values for which the function suddenly becomes invalid.
It's different from functions involving square roots or divisions, which might limit which numbers can go into the function without creating mathematical no-nos like imaginary numbers or division by zero.
  • For quadratic functions, the domain is usually all real numbers because they can accept any \( x \) value without breaking any mathematical rules.
Real Numbers
Real numbers are pretty much all the numbers we usually deal with. This includes integers like -3, -2, -1, 0, 1, 2, and so forth, as well as fractions like 1/2, and irrational numbers such as the square root of 2 or \( \pi \).
They make up the foundation for most of the math we do from everyday calculations to advanced equations.
Real numbers do not include imaginary numbers, which are usually seen with expressions that involve \( i \), the square root of -1.
  • Whenever you hear "the domain is all real numbers," it means any number you can think of, from negative to positive, can go into the function.
  • The abbreviation for it is often \( (-鈭�, 鈭�) \).
Interval Notation
Interval notation is a really handy way to specify the domain of a function without writing too many words.
When we say the domain of a function is all real numbers, in interval notation, we write it as \((-鈭�, 鈭�)\).
This simply means that \( x \) can be any number from negative infinity to positive infinity.
  • The parentheses \(()\) are used to show that infinity is not a specific, reachable number, but rather, the domain is open ended on both sides.
  • If a function has limits, like \( x \) can only be from 1 to 10, it鈥檇 be written as \([1, 10]\), where the brackets \([]\) indicate that the endpoints are included in the domain.

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