Problem 4 - College Algebra with Modeling and Visualization

Question

Determine whether \(f\) is a rational function and state its domain. $$ f(x)=\frac{x^{2}+1}{\sqrt{x-8}} $$

Step-by-Step Solution

Answer

\( f(x) \) is not a rational function. The domain is \([8, \infty)\).

Step 1: Identify the Function Type

A rational function is in the form \( \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. The function given is \( f(x)=\frac{x^{2}+1}{\sqrt{x-8}} \). Here, \( x^2+1 \) is a polynomial, but \( \sqrt{x-8} \) is not a polynomial (due to the square root). Therefore, \( f(x) \) is **not** a rational function.

Step 2: Determine the Domain of the Function

The domain of \( f(x) \) is determined by the values that make the denominator non-zero and defined. For \( \sqrt{x-8} \) in the denominator, \( x-8 \geq 0 \) to make the expression under the square root non-negative. Therefore, \( x \geq 8 \). The domain is \( [8, \infty) \).

Key Concepts

Domain of a Function

The domain of a function is all about finding the set of all possible input values (often represented as \(x\)) that your function can handle without running into any trouble. Knowing the domain of a function helps ensure that calculations are valid and avoid errors, like division by zero or trying to take the square root of a negative number.

For the given function in the exercise, \( f(x)=\frac{x^{2}+1}{\sqrt{x-8}} \), we focus on the denominator because that's where trouble might arise. Here, \( \sqrt{x-8} \) needs to be evaluated carefully. The expression under the square root, \(x - 8\), must be non-negative for \( \sqrt{x-8} \) to work. Hence, \( x - 8 \geq 0 \) leads us to \( x \geq 8 \).

Thus, the domain of the function is all real numbers \(x\) such that \(x\) is greater than or equal to 8. We write this as the interval \([8, \infty)\).

Polynomials

Polynomials are mathematical expressions involving a sum of powers of variables, like \(x\), with coefficients. They look like this: \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). The powers (or exponents) are always non-negative integers, and the coefficients are real numbers.

A few things to know about polynomials:

• They can have constants (e.g., \(5\)), variables with non-negative integer exponents (e.g., \(x^2\)), or both.
• The degree of a polynomial is the highest exponent of its variable.
• Polynomials are continuous and smooth, with no breaks or sharp corners.

In the exercise, the numerator \(x^2 + 1\) is a polynomial. It fits the definition perfectly as it consists of terms with \(x\) raised to powers, specifically \(x^2\) and the constant \(1\). This contrasts with the denominator, which involves a square root, a feature not allowed in polynomials.

Function Analysis

Analyzing a function involves understanding its behavior and characteristics throughout its domain. This includes examining the type of function, its domain, range, intercepts, continuity, and more. Here, we are particularly interested in identifying if the function is a "rational function."

A rational function is formed by the ratio of two polynomials: \( \frac{p(x)}{q(x)} \). To qualify as a rational function, both \( p(x) \) and \( q(x) \) must strictly be polynomials.

In the exercise, \( f(x)=\frac{x^{2}+1}{\sqrt{x-8}} \) doesn't meet this criterion because although the numerator \(x^2 + 1\) is a polynomial, the denominator \(\sqrt{x-8}\) is not since polynomials cannot include square roots. This is why \( f(x) \) is not a rational function. It’s this nuanced distinction that’s crucial in function analysis. Understanding these subtleties helps in correctly classifying and working with functions in mathematics.