Problem 80 - 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. $$f(x)=5 x-6$$

Step-by-Step Solution

Answer

The simplified difference quotient is 5.

Step 1: Compute f(x + h)

Given \(f(x) = 5x - 6\), substitute \(x + h\) for \(x\):
\(f(x+h) = 5(x+h) - 6 = 5x + 5h - 6\)

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

\(f(x+h) - f(x) = (5x + 5h - 6) - (5x - 6) = 5x + 5h - 6 - 5x + 6 = 5h\)

Step 3: Divide by h

\(\frac{f(x+h) - f(x)}{h} = \frac{5h}{h} = 5\)

The simplified difference quotient is \(\boxed{5}\). This is expected since \(f(x) = 5x - 6\) is linear with slope 5.

Key Concepts

Function Evaluation

Evaluating a function, especially in the context of the difference quotient, involves substituting a specific expression into the function itself. In our exercise, we start with the function \(f(x) = 5x - 6\). To evaluate the function at \(x + h\), substitute \(x + h\) for \(x\) in the expression:

• Original function: \(f(x) = 5x - 6\)
• Substitute \(x + h\) into the function: \(f(x+h) = 5(x+h) - 6\)

This substitution step is crucial for calculating the difference quotient, setting the stage for further algebraic manipulations to come. Evaluating functions correctly ensures that the rest of your computations will be accurate.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They are foundational elements in the study of algebra and pivotal for solving problems, such as finding the difference quotient. In the context of our problem, after substituting \(x + h\) into the function, we obtain an intermediate algebraic expression:

• Expression after substitution: \(f(x+h) = 5(x+h) - 6 = 5x + 5h - 6\)

Algebra involves manipulating these types of expressions to simplify them. Recognizing like terms and applying distributive properties are key methods you use in manipulating algebraic expressions. Understanding the underlying principles helps prevent common errors during transformations.

Simplification

Simplifying mathematical expressions is about reducing them to the most concise and clear form. This is especially important in the difference quotient, where simplicity will provide the clearest insight. After obtaining the expression for \(f(x+h)\), we continue by calculating \(f(x+h) - f(x)\):

• \(f(x+h) - f(x) = (5x + 5h - 6) - (5x - 6) = 5h\)

Once the subtraction is done, the difference \(5h\) is over \(h\):

• Difference quotient: \(\frac{5h}{h}\)
• Simplified form: \(5\) (assuming \(h eq 0\))

This final simplification step clarifies the relationship between the variables and numbers in the expression. Proper simplification practices, such as canceling out terms and verifying conditions like \(h eq 0\), ensure a straightforward and accurate result. These steps hone your algebraic skills and provide clear answers in calculus contexts.