Problem 37 - Algebra A Combined Function

Question

If \(f(x)=\frac{x^{2}+5}{x}\) and \(g(x)=\frac{x^{2}+2 x}{x+3},\) find each function value. $$ \text { Find the domain of } f(x) \text { . } $$

Step-by-Step Solution

Answer

The domain of \(f(x) = \frac{x^2 + 5}{x}\) is \((-\infty, 0) \cup (0, \infty)\).

Step 1: Identify Restriction on Domain for f(x)

The function is given as \(f(x) = \frac{x^2 + 5}{x}\). A fraction is undefined when its denominator is zero. Therefore, we need to find values where the denominator is zero. Here, the denominator is \(x\).

Step 2: Solve for Zero in the Denominator

Set the denominator equal to zero: \(x = 0\). Solve this equation to find that \(x\) cannot be zero.

Step 3: Determine the Domain

The domain of \(f(x)\) is all real numbers except \(x = 0\). In interval notation, this is \((-\infty, 0) \cup (0, \infty)\).

Key Concepts

Understanding Rational Functions

Rational functions involve the division of two polynomials. In mathematical terms, a function follows the form \(f(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials, and \(q(x)\) is not equal to zero. Some key features of rational functions include:

• The numerator can determine the overall behavior of the function in terms of roots and horizontal shifts.
• The denominator is crucial for identifying potential restrictions in the domain.
• These functions can exhibit asymptotic behavior, which means they approach a line but never actually reach it both in the horizontal and vertical directions.

Rational functions are essential in calculus and algebra, and they often appear in applications involving rates, such as speed or cost analysis.

Domain Restriction in Rational Functions

The domain of a function is the complete set of possible input values (x-values). For rational functions, determining the domain involves finding any x-values that would make the function undefined. Typically, this occurs when the denominator equals zero, as division by zero is undefined. For example, in the function \(f(x) = \frac{x^2 + 5}{x}\), the denominator \(x\) dictates the domain. To find any restrictions:

• Identify the denominator \(x\).
• Set the denominator equal to zero: \(x = 0\).
• Solve this to find that \(x\) cannot be zero.

Hence, the domain of \(f(x)\) includes all real numbers except zero, and is crucial for preventing undefined scenarios.

Handling Undefined Values

Undefined values in rational functions occur when the denominator equals zero. To ensure a rational function is correctly used, identifying and excluding these values from the domain is critical.When dealing with rational functions like \(f(x) = \frac{x^2 + 5}{x}\), finding where the denominator becomes zero tells us where the function is undefined.

• For \(f(x)\), set \(x = 0\). Since this makes the denominator zero, \(f(x)\) is undefined at \(x = 0\).

Once identified, these undefined values should be clearly noted, as failing to do so can lead to incorrect conclusions about the function's behavior.

Using Interval Notation

Interval notation is a concise way to express a set of numbers, especially useful for describing domains of functions. In interval notation, brackets and parentheses define the start and end of an interval.

• Parentheses \(( )\) are used when an endpoint is not included in the interval, often denoting points where a function is undefined.
• Brackets \([ ]\) indicate inclusivity of the endpoint.

For \(f(x) = \frac{x^2 + 5}{x}\), the domain excludes zero, leading to the domain expressed in interval notation as \((-\infty, 0) \cup (0, \infty)\). This notation effectively communicates that all real numbers except zero are part of the domain, providing a clear and concise representation of where the function is defined.