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Chapter 0: Problem 41

Find the domain of each function given below. $$f(x)=|x-4|$$

Short Answer

Expert verified
The domain of \( f(x) = |x - 4| \) is \( (-\infty, \infty) \).

Step by step solution

01

Understand the Function

The given function is \( f(x) = |x - 4| \). This is an absolute value function, which involves a transformation of the basic absolute value function \( |x| \).
02

Identify Domain Restrictions

To find the domain of \( f(x) \), determine if there are any restrictions on the values of \( x \) for which the function is defined. Absolute value functions are defined for all real numbers.
03

Write the Domain

Since there are no restrictions on \( x \) for the function \( f(x) = |x - 4| \), the domain includes all real numbers. Hence, the domain is \( (-\infty, \infty) \).

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Key Concepts

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

Absolute Value Functions
Absolute value functions form a critical part of algebra. They represent the distance of a value from zero on a number line. The most basic form is represented by \(|x|\).
The function \(f(x) = |x - 4|\) shifts the absolute value function horizontally. Instead of considering the distance of \(x\) from 0, we consider the distance of \(x\) from 4. This means the graph of the function \(|x - 4|\) is the same as that of \(|x|\), but moved to the right by 4 units.
Due to the properties of absolute value, this function ensures we always get a non-negative result for any real number input.
Function Domain
The domain of a function is the set of all possible input values (\(x\)) for which the function is defined. For many functions, there can be restrictions that limit the domain. Common restrictions arise from:
  • Values that make a denominator zero (undefined).
  • Values that result in the square root of a negative number.

However, the absolute value function doesn't face these issues. It converts any input value, be it positive or negative, into a non-negative output. Hence, it is defined for all real numbers. For the function \(f(x) = |x - 4|\), we perform a horizontal shift, but this does not affect its domain. As a result, you can plug in any real number into \(f(x)\) without causing any mathematical problems.
Real Numbers
Real numbers are all the numbers on the number line. They include rational numbers (like fractions and whole numbers) and irrational numbers (like square roots of non-perfect squares and \(\pi\)).
When we say a function is defined for all real numbers, we mean it works for any number you can think of, whether it's 1, -5, 0.75, or even \(\sqrt{2}\). Absolute value functions, like \(f(x) = |x - 4|\), fall into this category.
This is why the domain of the function \(-\infty, \infty \) means that the function can take any real number as an input.

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