Problem 34 - College Algebra

Question

In Exercises \(33-44\), find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=7 x$$

Step-by-Step Solution

Answer

The simplified difference quotient for the function \(f(x)=7x\) is 7.

Step 1: Substitute the function into the difference quotient

Using \(f(x) = 7x\), substitute into the difference quotient formula to get \(\frac{f(x+h)-f(x)}{h} = \frac{7(x+h) - 7x}{h}\)

Step 2: Simplify the numerator

By the distributive property, we can further simplify the fraction as follows: \(\frac{7x + 7h - 7x}{h}\). After canceling terms in the numerator, we get \(\frac{7h}{h}\)

Step 3: Simplify the difference quotient

Finally, by canceling common factors in the numerator and denominator, we simplify the difference quotient to get 7

Key Concepts

Simplifying Expressions

To understand algebraic problems, simplifying expressions is essential. It allows us to take a complex mathematical expression and make it more manageable. In the context of the difference quotient, simplifying the expression is crucial to reduce it to its most basic form.

For instance, when given the expression \[ \frac{7(x+h) - 7x}{h} \], simplifying involves distributing the 7 to both terms inside the parentheses, which results in \(7x + 7h - 7x)\). Afterward, we notice that \(7x\) and \( -7x\) are like terms and can be canceled out, leaving us with \(7h\). These steps are not just arbitrary; they utilize algebraic techniques such as the distributive property and combining like terms. These form the backbone of simplification processes and are widely applicable in different areas of mathematics.

When simplifying expressions, be sure to:

• Distibute any coefficients to terms inside parentheses.
• Combine like terms by adding or subtracting them accordingly.
• Cancel terms out if they appear both in the numerator and denominator of fraction.

Through these methods, we make expressions less intimidating and create a clear path to the solution.

Algebraic Techniques

Algebraic techniques are the tools that help us manipulate algebraic expressions to move from complexity to simplicity. These include various properties of numbers and operations, such as the distributive property, which we use when expanding expressions like \(7(x+h)\). When we multiply 7 by each term inside the parentheses, we are using this property to rewrite the expression without altering its value.

Another technique is term cancellation, where we subtract or add terms to eliminate them, especially when they appear with opposite signs. In the example, after expanding, we immediately see that \(7x\) and \( -7x\) cancel each other out because of their opposing signs.

• Always look for terms that can simplify directly, like numerical coefficients and constants.
• Keep track of signs while combining terms to avoid errors.
• Be methodical, and reduce expressions one step at a time for clarity.

These algebraic techniques form the foundation for understanding and solving more complex mathematical problems.

Function Notation

Function notation is a way to denote functions in a clear and concise manner, using symbols rather than words. In our difference quotient example, \(f(x) = 7x\) represents a function where the value of \(f(x)\) is determined by multiplying the input \(x\) by 7.

When we see \(f(x+h)\), it signifies the value of the function when \(x\) is replaced by \(x+h\). This change of input is crucial for evaluating functions at different points, which is why function notation is so powerful.

• Recognize that \(f(x)\) designates a function with \(x\) as the variable.
• Interpreting \(f(x+h)\) as the function's value after the input \(x\) is increased by \(h\).
• Understand that function notation helps in identifying the relationship between various function values.

With a firm grasp of function notation, you can move between different function values with ease, leading to a deeper understanding of functions and their properties.