Problem 6 - College Algebra with Modeling and Visualization

Question

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=\frac{4}{x}+1 $$

Step-by-Step Solution

Answer

Yes, \(f\) is a rational function with domain \(x \in \mathbb{R}, x \neq 0\).

Step 1: Identify if f is a Rational Function

A rational function is defined as a function that is the ratio of two polynomials. The given function is \(f(x) = \frac{4}{x} + 1\). Here, \(\frac{4}{x}\) can be rewritten as \(\frac{4}{x^1}\), where both the numerator and denominator are polynomials. Thus, this function is a rational function.

Step 2: Determine the Domain of the Function

The domain of a rational function includes all real numbers except where the denominator equals zero. For the function \(f(x) = \frac{4}{x} + 1\), the denominator is \(x\). Setting the denominator equal to zero gives \(x=0\). Hence, the function is undefined at \(x=0\). Therefore, the domain of \(f(x)\) is all real numbers except \(x = 0\), or \(x \in \mathbb{R}, x eq 0\).

Key Concepts

Domain of a Function

In mathematics, the domain of a function is the set of all possible input values (often represented as "x") for which the function is defined and outputs a real number. When considering functions, especially rational functions, it is crucial to identify these input values because some might be restricted due to division by zero errors.

• For most functions, such as linear or quadratic, the domain is usually all real numbers because there are no restrictions on x.
• However, when dealing with rational functions (ratios of polynomials), you must exclude any x-values that make the denominator zero to avoid undefined operations.

For example, in the function \( f(x) = \frac{4}{x} + 1 \), the domain is all real numbers except where x equals zero, since division by zero is not permitted in mathematics.

Polynomials

Polynomials are mathematical expressions involving a sum of powers of x with constant coefficients. They form the foundation for many areas of mathematics and include terms such as constants, variables, and exponents.

• Simple examples include expressions like \( 2x^2 + 3x + 4 \) or \( x^5 + 7x^3 - x + 9 \).
• Polynomials can have various degrees, determined by the highest power of x present in the expression.

Polynomials make up both the numerator and the denominator in rational functions. In the case of \( f(x) = \frac{4}{x} + 1 \), the numerator is the constant polynomial 4, which can also be viewed as \( 4x^0 \), and the denominator is a simple polynomial \( x^1 \). This construction is what qualifies the entire expression as a rational function.

Undefined Values

Undefined values occur in a function when the mathematical operation cannot be completed, often due to division by zero. In rational functions, these are the x-values that make the denominator zero, causing the function to lose its definition at those points.

• To find these undefined values, set the denominator of the rational function to zero and solve for x.
• These are the values excluded from the domain, ensuring the function remains valid and defined elsewhere.

For the function \( f(x) = \frac{4}{x} + 1 \), the denominator \( x \) becomes zero when x is zero. Hence, the function is undefined at \( x = 0 \), making it crucial to exclude this from the domain to maintain the function's validity.