Problem 56 - College Algebra

Question

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

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

Start by substituting \(f(x + h)\) and \(f(x)\) into the difference quotient formula. This will look like this: \(\frac{f(x+h)-f(x)}{h} = \frac{7(x + h) - 7x}{h}\)

Step 2: Simplifying the Expression

Now simplify the expression in the numerator: \(7x + 7h - 7x\). The terms \(7x\) and \(-7x\) cancel each other out, leaving us with \(7h\). Carry out division by h (which is the denominator) to get: \(\frac{7h}{h}\)

Step 3: Final Simplification

The final step is to simplify the difference quotient fully. The \(h\) in the numerator and the denominator cancel each other, leaving us with \(7\). This is the final answer.

Key Concepts

Functions

A function is a special relationship between two sets, typically denoted as inputs and outputs. In this relationship, each input is related to exactly one output. This is what makes functions so reliable and usable in mathematics.
For example, consider the function \( f(x) = 7x \). Here, the input is \( x \), and the function multiples \( x \) by 7 to get the output. So, if you input 1, the output is 7. Put in 2, and you get 14, simply by multiplying 2 by 7. An important property of functions is their repeatability and predictability: knowing the rule gives you confidence in the output from any input.
Functions are foundational in mathematics, allowing you to describe real-world situations with equations and make accurate predictions or assessments.

Simplification

Simplification involves reducing expressions into simpler or more manageable forms. It is vital in making complex problems more approachable, especially in algebra. Simplifying often involves:

• Combining like terms for efficiency.
• Canceling terms when they are equal but opposite, such as \( x - x = 0 \).
• Using arithmetic operations to streamline expressions.

Simplification also plays a major role in the step-by-step solution of the difference quotient. Initially, the difference quotient expression \( \frac{7(x+h) - 7x}{h} \) might look complex, but through simplification:

• Terms like \( 7x - 7x \) cancel, clarifying the expression.
• Reducing \( \frac{7h}{h} \) to 7 by canceling \( h \).

By making an equation simpler, it reveals the core result or relationship more clearly, often making it easier for further analysis or application.

Difference Quotient Formula

The difference quotient is an essential concept for understanding how functions change; it measures the rate of change and is the basis for derivatives in calculus. The formula is written as:

\[\frac{f(x+h) - f(x)}{h}, \text{ where } h eq 0\]
It represents the average rate of change of the function \( f \) over the interval \( x \) to \( x+h \).
To compute it:

• First determine \( f(x + h) \), which is the function applied at \( x + h \).
• Next, identify \( f(x) \), which is your original function at \( x \).
• Subtract these, \( f(x + h) - f(x) \) and then divide by \( h \).

This provides a way to approximate how the function behaves between two points. Simplifying the result then gives a clearer picture of the function's main trend over a small interval.

College Algebra

College algebra is an advanced level of algebra that builds on high school algebra. It delves deeper into understanding functions, equations, and systems. Topics covered usually include:

• Polynomials, rational expressions, and complex numbers.
• Exponential and logarithmic functions for growth models.
• Analyzing different types of functions and graphs.
• Foundations for calculus through concepts such as limits and the difference quotient.

In this context, the exercise of finding and simplifying the difference quotient represents an important aspect of college algebra. It combines function evaluation with simplification processes, preparing students for the more sophisticated calculus ideas. Mastery of such concepts enhances analytical skills and problem-solving capabilities critical for success in more advanced classes and various scientific or engineering fields.