Problem 79 - A Graphical Approach to College Algebra

Question

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) ) for each function \(f\). Simplify completely. $$f(x)=4 x+3$$

Step-by-Step Solution

Answer

The simplified difference quotient is 4.

Step 1: Substitute Function into the Difference Quotient

First, substitute the given function \( f(x) = 4x + 3 \) into the difference quotient formula. The difference quotient is given by \( \frac{f(x+h) - f(x)}{h} \). We will need to find \( f(x+h) \) first by replacing \( x \) with \( x+h \) in the expression for \( f(x) \).

Step 2: Compute \( f(x+h) \)

Calculate \( f(x+h) \) by replacing \( x \) with \( x+h \) in the function \( f(x) = 4x + 3 \). This yields \( f(x+h) = 4(x+h) + 3 \). Expand this expression to get \( 4x + 4h + 3 \).

Step 3: Set Up the Difference Quotient

Substitute \( f(x+h) = 4x + 4h + 3 \) and \( f(x) = 4x + 3 \) into the difference quotient formula: \( \frac{(4x + 4h + 3) - (4x + 3)}{h} \).

Step 4: Simplify the Numerator

Simplify the numerator \((4x + 4h + 3) - (4x + 3)\) by distributing and combining like terms. This results in \(4h\).

Step 5: Simplify the Difference Quotient

Insert the simplified numerator back into the difference quotient: \( \frac{4h}{h} \). Since \(h eq 0\), we can cancel \(h\) in the numerator and denominator, resulting in a simplified form: \(4\).

Key Concepts

Function Substitution

Function substitution is a fundamental step when dealing with difference quotients. It involves replacing the variable in a function with another expression. This step sets the stage for further calculations. For instance, given the function \( f(x) = 4x + 3 \) and the expression \( x+h \), substituting \( x+h \) into the function requires replacing every instance of \( x \) with \( x+h \).

• Start by noting the given function: \( f(x) = 4x + 3 \)
• Substitute \( x+h \) for \( x \) to get \( f(x+h) = 4(x+h) + 3 \)
• Distribute to obtain the expanded form: \( f(x+h) = 4x + 4h + 3 \)

This process prepares the function for use in the expression \( \frac{f(x+h) - f(x)}{h} \). Understanding substitution is crucial as it ensures the correct expression is used in upcoming simplifications.

Simplifying Expressions

Simplifying expressions is an important skill in mathematics, especially in calculus problems like finding the difference quotient. After substituting variables, as seen with \( f(x+h) \), the next step is to simplify the expression within the difference quotient.Let's consider the difference quotient setup:

• Start with \( \frac{(4x + 4h + 3) - (4x + 3)}{h} \)
• The goal is to simplify the numerator: \((4x + 4h + 3) - (4x + 3)\)
• Subtract the terms: \(4x + 4h + 3 - 4x - 3 \)

This subtraction step helps in removing identical terms and prepares the expression for further simplifications. Keeping each calculation distinct and identifying terms that "cancel out" simplifies solving.

Combining Like Terms

Combining like terms simplifies the expression and makes the computation easier. In the context of the difference quotient, this means carefully managing terms to isolate those affected by the variable \( h \).Consider the expression gotten from the simplified numerator:

• Focus on: \(4h\)
• Notice how terms like \(4x\) and constant \(3\) have been eliminated through subtraction
• This leaves \(4h\) as the sole term in the numerator

After combining like terms, the remaining expression is \( \frac{4h}{h} \). Since \( h eq 0 \), you can cancel \( h \) from the numerator and denominator, simplifying the expression to \( 4 \). This concise result represents the simplification and highlights the importance of combining terms to achieve a clearer, final answer.