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Chapter 3: Problem 10

Find the domain of each rational function. $$f(x)=\frac{4-x}{x+2}$$

Short Answer

Expert verified
The domain of the function is all real numbers except x = -2, which is written in interval notation as \[ (-\infty, -2) \cup (-2, \infty) \].

Step by step solution

01

- Identify the Rational Function

The given function is defined as: \[ f(x) = \frac{4 - x}{x + 2} \] A rational function is a fraction where both the numerator and the denominator are polynomials.
02

- Determine the Denominator's Domain

To find the domain of the rational function, identify the values of x that make the denominator equal to zero. The denominator of the given function is: \[ x + 2 \]
03

- Solve for Zero in the Denominator

Set the denominator equal to zero and solve for x: \[ x + 2 = 0 \] Solving this equation gives: \[ x = -2 \]
04

- Exclude This Value from the Domain

Since a rational function is undefined when the denominator is zero, exclude \( x = -2 \) from the domain. Therefore, the domain of \( f(x) \) is all real numbers except \( x = -2 \).
05

- Write the Domain in Interval Notation

The domain of the function in interval notation is: \[ (-\infty, -2) \cup (-2, \infty) \]

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Key Concepts

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

Rational Functions
Rational functions are fractions where both the numerator and the denominator are polynomials. Polynomials are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In the case of our example, the numerator is 4 - x, and the denominator is x + 2. Understanding rational functions is crucial because they often show up in algebra and calculus. The key thing to remember here is that rational functions can be undefined at certain points, which we will explore further.
Polynomials
Polynomials are expressions formed by summing multiples of powers of a variable. Polynomials include terms like x, x^2, and x^3, and can appear in many different forms. In our example, the numerator (4 - x) and the denominator (x + 2) are both simple polynomials. The degree of a polynomial is the highest power of the variable. For instance, x and x + 2 are polynomials of degree 1. Understanding this helps when simplifying or factoring them for rational functions.
Interval Notation
Interval notation is a way of describing the set of all x-values that satisfy a given condition. In our example, we've determined that the function f(x) = (4 - x)/(x + 2) is undefined when x = -2. This means that our domain, which consists of all possible x-values except -2, can be expressed in interval notation. We use parentheses to indicate that a value is not included in the interval. So, the domain is written as (-鈭�, -2) 鈭� (-2, 鈭�). This way, we clearly communicate that -2 is excluded from the range of acceptable values.
Undefined Values
Undefined values in rational functions occur when the denominator equals zero because division by zero is not possible. In our specific example, the denominator is x + 2. Setting the denominator equal to zero, we solve for x: x + 2 = 0, which gives us x = -2. This tells us that the function f(x) = (4 - x)/(x + 2) is not defined at x = -2, making it an undefined value. As a result, x = -2 must be excluded from the domain of the function. This process of identifying undefined values ensures that we correctly determine where the function can and cannot be evaluated.

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