Problem 96 - College Algebra

Question

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has no vertical. horizontal, or slant asymptotes, and no \(x\) -intercepts.

Step-by-Step Solution

Answer

A rational function that meets the conditions stated in the exercise is \(f(x) = \frac{a}{b}\), where \(a\) and \(b\) are any constants and \(b\neq 0\). An example is function \(f(x) = 1\) (where \(a = 1\) and \(b = 1\)), which is a constant function with no x-intercepts and no asymptotes.

Step 1: Define Polynomials \(p(x)\) and \(q(x)\)

Define \(p(x)\) and \(q(x)\) as polynomials such that their degree is equal and the function \(f(x)\) does not intercept with x-axis or any asymptotes. An example of such polynomials are constant polynomials where there are no \(x\)'s involved and the degree is 0. Therefore, define \(p(x) = a\) and \(q(x) = b\) where \(a\) and \(b\) are constants and \(b\neq 0\).

Step 2: Substitute Polynomials into the Function

Substitute \(p(x)\) and \(q(x)\) into the function \(f(x)\). This results in \(f(x)=\frac{a}{b}\).

Step 3: Function Verification

It's important to note that depending upon the values of \(a\) and \(b\), the function \(f(x)=\frac{a}{b}\) can represent different functions. For example, if \(a = 1\) and \(b = 1\), the function becomes \(f(x) = 1\), which is a constant function. This function does not intersect with the x-axis (as there are no x-intercepts) and does not have any asymptotes since the function doesn't approach any value as \(x\) goes to infinity or negative infinity.

Key Concepts

Polynomials

In mathematics, polynomials are expressions composed of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They form the backbone of algebra and come in various shapes and forms. The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial \(3x^2 + 2x + 1\), the degree is 2 due to the highest exponent of the variable, which is 2 in \(3x^2\). However, in many scenarios, we encounter constant polynomials where there are no variables, only numbers.When we talk about rational functions like \(f(x) = \frac{p(x)}{q(x)}\), the polynomials \(p(x)\) (the numerator) and \(q(x)\) (the denominator) dictate the behavior of the function. Constant polynomials, where both \(p(x)\) and \(q(x)\) are constants, create a situation where the function is a constant ratio of these two values. This is what makes rational functions with constant polynomials truly unique because they manifest as constant functions without variables.

Asymptotes

An asymptote is a line that a graph of a function approaches but never actually touches. There are different types of asymptotes:

• Vertical Asymptotes: These occur when the function's denominator is zero, causing the function to grow infinitely large.
• Horizontal Asymptotes: These are found in rational functions when the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial.
• Slant (or Oblique) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator.

If a rational function \(f(x)\) doesn't have any asymptotes, it suggests that neither the vertical division caused by the denominator nor the horizontal or slant balancing happens. In such a case, both the numerator and the denominator are constants, indicating that the division yields another constant without infinity approaching behavior.

Constant Function

A constant function is one of the simplest functions in mathematics. It produces the same output value regardless of the input. In other words, it graphs as a horizontal line in the Cartesian plane. When a rational function like \(f(x)=\frac{a}{b}\) is defined using constant polynomials, \(a\) and \(b\) are non-zero constants, and the function remains constant for all values of \(x\). This means that there is no change, no curve, and no intercepts.For example, if \(a = 1\) and \(b = 1\), then \(f(x) = 1\) for all values of \(x\). In this way, constant functions represent stable, unchanging relationships where the graph is just a straight horizontal line.

X-Intercepts

An \(x\)-intercept is a point where a graph crosses the \(x\)-axis, meaning the function’s output value is zero at that point. In terms of a rational function \(f(x)=\frac{p(x)}{q(x)}\), an \(x\)-intercept exists if \(p(x)=0\) while \(q(x)eq0\). This means if the numerator can be factored to produce a zero, the rational function will intercept the \(x\)-axis at these points. However, if the function is defined such that \(p(x)=\text{constant}\) and \(q(x)=\text{constant}\), the numerator is never zero and therefore, the function can never cross the \(x\)-axis. Such a function will keep constant and maintain a distance from zero, affirming that it does not have an x-intercept.