Problem 117 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

Step-by-Step Solution

Answer

The statement makes sense. The difference quotient is always zero for a constant function since the rate of change is zero, as the constant function does not change over time.

Step 1: Define the difference quotient

The difference quotient is defined as \((f(h) - f(0))/h\). This is a measure of the average rate of change of a function between any two points.

Step 2: Apply the difference quotient to a constant function

Let's see what happens when we apply the difference quotient to a constant function \(f(x) = c\), where \(c\) is any constant. Now \(f(h) = c\) and \(f(0) = c\), so the difference quotient is \((c-c)/h = 0/h\).

Step 3: Evaluate the difference quotient

Zero divided by any number \(h\) is zero, so the difference quotient is indeed always zero for a constant function.

Key Concepts

Constant Function

When dealing with functions, a constant function is one of the simplest types imaginable. As the name suggests, a constant function is represented mathematically as \(f(x) = c\), where \(c\) is a constant value, meaning it does not change no matter what the input value \(x\) is.

Imagine drawing a constant function on a graph; you'll end up with a straight, horizontal line, because the output never varies—it's always \(c\). This consistency across input values means that if you calculate the difference quotient of a constant function, as revealed in the textbook exercise, you will indeed find that it's zero. There's no change to measure when comparing any two points on the graph because they all lie on the same horizontal line with the same y-value \(c\). This is a reflection of the fact that the average rate of change for a constant function is zero, as it simply does not change.

Average Rate of Change

The average rate of change is a fundamental concept in mathematics that describes how much a function changes on average over a certain interval. Think about it as the 'slope' that you might recall when learning about linear functions, but applied more generally.

The average rate of change is calculated by the difference quotient, which is defined as the change in the function's value over the change in the function's input. In algebraic terms, if you have a function \(f(x)\), the average rate of change from \(x=a\) to \(x=b\) is defined as \(\frac{f(b)-f(a)}{b-a}\). When you have a constant function, the numerator of this fraction is always zero because \(f(b)\) and \(f(a)\) are the same. Hence, the average rate of change for a constant function is always zero — it does not grow, diminish, or oscillate as the input changes.

Algebraic Concepts

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. The algebraic concepts involve manipulating these symbols to solve problems, analyze patterns, and understand relationships between variables. In the context of the exercise, we utilize algebraic manipulation to understand the behavior of the difference quotient for a constant function.

Applying algebraic principles, we can establish that the difference quotient for a constant function simplifies to zero, as proven through the exercise's solution. This algebraic exploration is an excellent example of how abstract symbols and simple calculations can lead to a concrete understanding of a function's behavior. It's also a reminder of how the foundational concepts of algebra underpin much of the work we do in higher-level mathematics, offering tools to analyze and solve a broad array of problems.