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Chapter 13: Problem 14

Find the domain. $$ f(x)=\frac{4-x}{2} $$

Short Answer

Expert verified
Answer: The domain of the function $$f(x) = \frac{4-x}{2}$$ is all real numbers, which can be represented in interval notation as $$(-\infty, \infty)$$.

Step by step solution

01

Look for Restrictions for the Function

In this function, there are no square roots, logarithms, or other functions that would put restrictions on the values of x. The only thing that could be a problem is if the denominator is equal to zero. However, the denominator of the given function is 2, which is not equal to zero.
02

Determine the Domain of the Function

Since there are no restrictions on the values of x, the function is defined for all real numbers. Thus, the domain of the function $$f(x) = \frac{4-x}{2}$$ is all real numbers.
03

Write the Domain in Interval Notation

To express the domain of the function in interval notation, we can use the following notation: $$(-\infty, \infty)$$ This represents all real numbers, which is the domain of the given function. So, the domain of the function $$f(x) = \frac{4-x}{2}$$ is $$(-\infty, \infty)$$.

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Key Concepts

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

Real Numbers
Real numbers encompass all the numbers you can imagine measuring on a continuous number line. They include both rational and irrational numbers.
  • Rational numbers are those numbers that can be expressed as the quotient of two integers (e.g., 1/2, 5, -3).
  • Irrational numbers are those that cannot be written as a simple fraction (e.g., 鈭�2, 蟺).
In the context of functions, the domain often consists of real numbers, unless there is something in the function itself that prevents certain values of 'x' from being used. In the given exercise, we need to determine where the function is "happy" or defined.The function \(f(x) = \frac{4-x}{2}\) can take all values of 'x' that are real numbers since subtracting a real number and dividing by 2 doesn't exclude any x-values. Hence, the domain is all real numbers.
Interval Notation
Interval notation is a way of representing a range of numbers in a concise manner. It is especially useful when we want to indicate the domain of a function that includes all or part of the real numbers. For instance, \((-\infty, \infty)\) covers all real numbers. It means the interval stretches from negative infinity to positive infinity, including everything in-between. Here, parentheses are used to denote that the endpoints are not included.
  • "(" and ")" imply that the endpoints are not part of the domain.
  • "[" and "]" indicate that the endpoints are included in the interval.
The function \(f(x) = \frac{4-x}{2}\) is defined for all real numbers, so its domain is written as \((-\infty, \infty)\) in interval notation.
Function Restrictions
In mathematics, some functions have restrictions that limit the set of input values they can accept, due to operations that are not defined for all real numbers. Common restrictions arise from:
  • **Division by zero**: Division by zero is undefined, so any value that makes a denominator zero is excluded from the domain.
  • **Square roots of negative numbers**: These are not included if the function is real-valued because they result in complex numbers.
  • **Logarithms of non-positive numbers**: These are undefined because the log of zero or a negative number is not defined in the set of real numbers.
In the given function \(f(x) = \frac{4-x}{2}\), there are no functions such as square roots or logarithms involved. The denominator is a constant number 2, which is not zero, so there are no restrictions. Therefore, the domain includes all real numbers.

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Most popular questions from this chapter

Put the rational expressions into quotient form, identify any horizontal or slant asymptotes, and sketch the graph. $$ \frac{x^{2}-5 x+7}{x-2} $$

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