Question

Find the domain of each function. \(f(x)=\frac{-x}{4}\)

Short answer

The domain is \( (-\infty, \infty) \).

Step-by-step solution

Understand the Function

The given function is a linear function expressed as \( f(x) = \frac{-x}{4} \). This is a simple linear expression.

Identify Restrictions for the Domain

For a linear function like \( f(x) = \frac{-x}{4} \), the domain includes all real numbers because a linear function is defined for every real number.

Express the Domain

Since there are no restrictions on x, the domain is all real numbers. In interval notation, this is represented as \( (-\infty, \infty) \).

Key concepts

linear functions

A linear function is one of the simpler types of functions in mathematics. Its general form is represented as \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the given exercise, we have \( f(x) = -\frac{x}{4} \). This means our slope \( m \) is -\( \frac{1}{4} \), and our y-intercept \( b \) is 0. Linear functions graph as straight lines on the coordinate plane. They have the key property that each input \( x \) is mapped to exactly one output \( f(x) \). This consistent relationship makes linear functions easy to work with and a fundamental starting point in algebra and calculus.

function domain

The domain of a function is the set of all possible input values, usually represented by \( x \) in the function \( f(x) \). For the given linear function, the domain is all real numbers. This is because there are no restrictions preventing any real number from being substituted in for \( x \). To determine the domain, consider whether there are any constraints like division by zero or taking the square root of a negative number. Since none of these issues apply to our linear function \( f(x) = -\frac{x}{4} \), its domain is simply all real numbers.

interval notation

Interval notation is a way to represent subsets of the real numbers. It is especially useful for describing the domain and range of functions. For a linear function like \( f(x) = -\frac{x}{4} \), where the domain is all real numbers, interval notation is used to express this range. To write it, we use the symbols for infinity: \( -\infty \) and \( \infty \). So, the domain of our function in interval notation is written as \( (-\infty, \infty) \). This notation tells us that \( x \) can be any real number, extending infinitely in both the positive and negative directions without any breaks or gaps.