Problem 1 - Beginning and Intermediate Algebra

Question

Find the domain of each rational expression. See Example 1. $$ f(x)=\frac{5 x-7}{4} $$

Step-by-Step Solution

Answer

The domain of \(f(x)=\frac{5x-7}{4}\) is all real numbers, \( (-\infty, \infty) \).

Step 1: Understand the Rational Expression

The function given is a rational expression. A rational expression is a fraction where both the numerator and the denominator are polynomials. In this case, the numerator is \(5x - 7\) and the denominator is \(4\).

Step 2: Identify Restrictions

For a rational expression, the restrictions on the domain come from the denominator. A rational expression is undefined when its denominator is zero, as division by zero is undefined.

Step 3: Check the Denominator

Examine the denominator of the given rational expression, which is \(4\). Since \(4\) is a constant and not equal to zero, there are no values of \(x\) that will make the denominator zero.

Step 4: Determine the Domain

Since there are no values of \(x\) that make the denominator zero, the domain of the expression includes all real numbers.

Key Concepts

Rational Expressions

Rational expressions are essentially the fractions of algebra. These are mathematical phrases that represent the ratio of two polynomials. In the example provided, we see that our rational expression is \( \frac{5x - 7}{4} \). Here, the polynomial \(5x - 7\) forms the numerator, and \(4\), a constant, serves as the denominator.

• Numerator: This is the top part of the fraction, \(5x - 7\) in our example.
• Denominator: This part is crucial as it determines the expression’s domain. It's \(4\) in this case.

Understanding how these components work together is key in solving and simplifying rational expressions. By mastering them, you better appreciate the symbiotic relationship between polynomials in both the numerator and the denominator.

Polynomials

Polynomials are the backbone of rational expressions. They are algebraic expressions made up of variables, coefficients, and exponents that are non-negative integers. A single term of a polynomial is called a 'monomial', and more than one term results in a 'binomial' or 'trinomial', depending on how many terms exist.
In \(5x - 7\), \(5x\) is a monomial, and \(-7\) is another monomial, together making the polynomial \(5x - 7\). In the denominator, \(4\) stands alone as a constant polynomial.

• Constants like \(4\) might not fit your traditional image of polynomials, but they are indeed polynomials of degree zero.
• Analyzing these structures helps solve rational expressions by identifying terms that can be factored or canceled out.

Recognizing polynomials is an important skill within math; they frame the more complex equations you'll tackle.

Restrictions on Domain

When working with rational expressions, considering restrictions on their domain is crucial. The domain refers to all the possible input values (in terms of \(x\)) that result in a defined expression. Due to denominators, rational expressions have specific restrictions where the denominator cannot be zero.
The approach is straightforward: set the denominator equal to zero and solve for \(x\). However, for the expression \( \frac{5x - 7}{4} \), the denominator \(4\) is a constant and never zero, meaning there are no restrictions that will make it undefined.

• If a denominator was \(x + 2\), for example, you would solve \(x + 2 = 0\). Here, \(x = -2\) would make the expression undefined.
• In our example, since the denominator is never zero, the domain is all real numbers.

Thus, the result is that the domain extends across all numbers, showing that rational expressions involving constants in the denominator can have a much broader domain than those with variable polynomials.