Problem 55 - College Algebra

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=4 x$$

Step-by-Step Solution

Answer

The simplified difference quotient for the function \(f(x) = 4x\) is \(4\).

Step 1: Substitute the function and simplify numerator

Begin by substituting \(f(x)=4x\) and \(f(x+h)=4(x+h)\) into the difference quotient. Therefore, we have \(\frac{f(x+h)-f(x)}{h} = \frac{4(x+h) -4x}{h}\). Then, simplify the numerator.

Step 2: Expanding the numerator

Next, we would expand the numerator. This would give \(\frac{4x + 4h -4x}{h}\).

Step 3: Cancelling expression in the numerator

The \(4x\) and \(-4x\) cancel out in the numerator leaving us with \(\frac{4h}{h}\).

Step 4: Simplifying the quotient

We now have \(\frac{4h}{h}\) left. We divide \(4h\) by \(h\) and get \(4\).

Key Concepts

Simplifying Algebraic Expressions

Grasping the concept of simplifying algebraic expressions is fundamental to understanding and solving more complex algebraic problems. Take for example the expression in the numerator of the difference quotient, \(4(x+h) - 4x\). Here, we want to combine like terms to make the expression as straightforward as possible.

When we're dealing with an algebraic expression involving a common factor across several terms, we can make our task easier by first identifying the factor and then distributing it across the terms. This process is called factoring. In our case, the common factor is 4, and we distribute it across the terms inside the parentheses, resulting in \(4x + 4h\), and then we subtract \(4x\). After distributing, we identify and combine like terms, which are the terms involving \(4x\) in this instance. They cancel each other out, leaving us with \(4h\). This simplification makes it easier to understand what we're working with and prepares us for the next step: dividing the whole expression by \(h\), which is the concept of function evaluation.

Function Evaluation

The skill of function evaluation takes center stage in the application of the difference quotient. Once we have simplified our expression to \(\frac{4h}{h}\), we must evaluate it by performing the operation indicated - division. This process of evaluation involves substituting specific values into a function and then simplifying the function accordingly.

Considering \(\frac{4h}{h}\), we observe that \(h\) is a common factor in both the numerator and the denominator. As per the rules of algebra, any non-zero number divided by itself is 1. Therefore, our function evaluation task here is simplifying this expression by dividing \(4h\) by \(h\), which effectively means removing the common factor \(h\) from both the numerator and the denominator to yield the constant 4. This final outcome is what the difference quotient is, evaluated for the function \(f(x) = 4x\).

Algebraic Techniques

Utilizing algebraic techniques is essential when working through expressions and equations to reach a simplified form or to evaluate a function effectively. The steps we've followed in solving the difference quotient involve several of these techniques including substitution, distribution, combining like terms, and cancelling out terms.

Substitution is where we started by replacing \(f(x)\) and \(f(x + h)\) with their respective algebraic forms in the initial difference quotient. Distribution came into play when we expanded \(4(x+h)\). We then combined like terms \(4x\) and \(−4x\), which simplifies to 0, and this is a critical step since it reduces the complexity of the expression. Finally, the cancellation of \(h\) in the numerator and denominator perfectly exemplifies the importance of recognizing common factors to simplify expressions. These algebraic techniques not only help in solving the specific problem at hand but also build a strong foundation for tackling more challenging problems in mathematics.