Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, find the average rate of change of each function on the interval specified. \(y=\frac{1}{x}\) on [1,3]

Step-by-Step Solution

Answer

The average rate of change is \(-\frac{1}{3}\).

Step 1: Identify the Formula for Average Rate of Change

The formula for the average rate of change of a function \(y\) over an interval \([a, b]\) is \(\frac{y(b) - y(a)}{b - a}\). Here, \(a = 1\) and \(b = 3\).

Step 2: Evaluate the Function at the Endpoints

First, evaluate the function \(y=\frac{1}{x}\) at the endpoints \(x = 1\) and \(x = 3\):- For \(x = 1\): \(y(1) = \frac{1}{1} = 1\).- For \(x = 3\): \(y(3) = \frac{1}{3}\).

Step 3: Compute the Average Rate of Change

Substitute the results from Step 2 into the average rate of change formula:\[\frac{y(3) - y(1)}{3 - 1} = \frac{\frac{1}{3} - 1}{3 - 1} = \frac{\frac{1}{3} - \frac{3}{3}}{2} = \frac{-\frac{2}{3}}{2}\]Simplify the fraction:\[= \frac{-2}{6} = -\frac{1}{3}\]

Step 4: Summarize the Result

The average rate of change of the function \(y = \frac{1}{x}\) over the interval \([1, 3]\) is \(-\frac{1}{3}\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output (or value) of a function for a specific input value. It's a fundamental skill that helps us understand how a function behaves at certain points. Essentially, you substitute the input value (usually denoted as "x") into the function and simplify to find the result.

To evaluate a function like the one given in this exercise, where the function is defined as \( y = \frac{1}{x} \), you perform the following steps:

• Identify the input values of interest, which in an average rate of change problem are usually the endpoints of the interval.
• Substitute the input values individually into the function. For example, if the input value is 1, replace every "x" in the equation with 1.
• Calculate the result. For \( y(1) = \frac{1}{1} = 1 \), the function yields the result of 1.

In problems involving the average rate of change, accurate function evaluation at each endpoint is key to determining how much the output changes across the interval.

Rational Functions

Rational functions are mathematical equations that can be expressed as the quotient of two polynomials. In simpler terms, they are functions in the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \). These functions often display interesting behaviors, such as asymptotes and discontinuities.

The function from our exercise, \( y = \frac{1}{x} \), is a simple rational function. Here, 1 is a constant polynomial, and the denominator is the polynomial \( x \). It's important to remember that the function is undefined where the denominator is zero, which means \( x = 0 \) is not in the domain of the function.

Rational functions can be characterized by:

• Their horizontal and vertical asymptotes. \( \frac{1}{x} \) has a vertical asymptote at \( x = 0 \).
• Their behavior as \( x \) approaches positive or negative infinity, which can offer insight into end-behavior of the graph.

Understanding these properties is essential for analyzing how rational functions behave over different intervals.

Interval Notation

Interval notation is a way of expressing subsets of the real number line, often used to denote the domain over which a function is evaluated or analyzed. It provides a concise and clear way of describing intervals, which is essential for understanding the scope of many mathematical problems.

In our example, the interval [1, 3] denotes all numbers from 1 to 3, inclusive. Here's a quick guide to interpreting interval notation:

• Brackets \([ \text{and} ]\) mean the endpoints are included, representing a closed interval.
• Parentheses \(( \text{and} )\) indicate the endpoints are not included, representing an open interval. For example, \((1, 3)\) would mean values greater than 1 and less than 3, but not including 1 or 3 themselves.
• Interval notation can be used to identify the domain or range of a function, or to define the scope over which calculations like averages are performed.

By understanding and accurately using interval notation, you can effectively communicate which parts of the real line are being considered in a problem.