Problem 37 - Introductory Algebra for College Students

Question

Find and simplify. \(\frac{f(x)-f(h)}{x-h}\) $$f(x)=6 x+7$$

Step-by-Step Solution

Answer

The simplified value is 6.

Step 1: Substitute function into derivative definition

We start the task by substituting the function \(f(x) = 6x + 7\) into the definition of the derivative: \(\frac{f(x)-f(h)}{x-h}\). We replace \(f(x)\) and \(f(h)\) with the values given in the function \(6x+7\) and \(6h+7\) respectively.

Step 2: Simplify

The formula would now look like this: \(\frac{[(6x+7) - (6h+7)]}{(x-h)}\) . Now, simplify by removing the parentheses, which gives us \(\frac{6x + 7 - 6h - 7}{x - h}\). This simplifies to \(\frac{6x - 6h}{x - h}\).

Step 3: Cancel out terms

Next, we cancel out terms. The \(6x-6h\) in the numerator is equivalent to \(6(x - h)\). Thus, the whole formula simplifies to \(\frac{6(x - h)}{x - h}\). The \((x - h)\) terms in the numerator and the denominator cancel each other out, leaving us with 6.

Key Concepts

Derivative

The derivative is a powerful tool in calculus used to measure how a function changes as its input changes. It's essentially the rate of change or the slope of a function at any given point.
To find a derivative, especially in introductory algebra, you often use the difference quotient formula: \(\frac{f(x+h) - f(x)}{h}\). This formula involves substituting specific function values into it and simplifying the results. It's instrumental in understanding deeper concepts such as motion, growth, and decay.

• The process starts with identifying the function. In our exercise, this function is \(f(x) = 6x + 7\).
• The function is then substituted into the derivative formula, replacing \(f(x)\) and \(f(h)\) with \(6x + 7\) and \(6h + 7\), respectively.

Understanding derivatives in simple linear functions like this one helps form a foundation for analyzing more complex functions later. By grasping this, you create a bridge to topics like optimization and the behavior of polynomial curves.

Simplification

Simplification in algebra aims to make an expression easier to work with by reducing it to its most basic form. It's a critical skill because it allows you to easily see relationships and patterns that might be obscured by more complicated expressions.
In the context of our problem, simplification involves three main steps:

• First, ensure all terms are correctly substituted into the equation: \(\frac{(6x+7)-(6h+7)}{x-h}\).
• Next, remove parentheses to combine like terms, leading us to \(\frac{6x + 7 - 6h - 7}{x - h}\).
• Finally, further reduce by cancelling any terms that behave similarly, transforming the expression seamlessly to \(\frac{6x - 6h}{x - h}\).

Each simplification step not only narrows down the expression but also spotlights the operation needed to solve or interpret the algebraic problem efficiently. It's like peeling away layers of an onion to get to the core, making the solution not just correct but elegant too.

Function Substitution

Function substitution is all about replacing specific variables or expressions within a function with other known values or functions. It's an essential algebraic process that allows us to evaluate and manipulate functions under different scenarios.
In the given problem, our primary task was to substitute \(f(x) = 6x + 7\) and \(f(h) = 6h + 7\) into the required formula. By doing so, we ensure that the operation we perform reflects the specific behavior of the function given.

• It's important during substitution to keep track of each alteration to maintain balance in the equation change, which can't be understated.
• This step precedes simplification, as it lays the groundwork for accurately processing expressions.

By mastering function substitution, you're preparing to seamlessly carry out operations on functions, making it easier to transition into more advanced topics such as integration and differential equations. It enhances analytical skills by allowing you to examine how changes in input values can shift or transform function outputs.