Problem 105 - Precalculus : Building Concepts and Connections

Question

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=3 x-1$$

Step-by-Step Solution

Answer

The difference quotient of the function \(f(x) = 3x - 1\) is \(3\).

Step 1: Substitution

Substitute \(x+h\) into the function \(f\), so \(f(x + h) = 3(x + h) - 1\). After doing that, distribute the \(3\) in \(3(x + h)\) and evaluate \(f(x+h) = 3x + 3h - 1\)

Step 2: Difference Quotient

Set up the difference quotient. Substitute \(f(x+h)\) and \(f(x)\) into the formula: \[\frac{f(x+h) - f(x)}{h} = \frac{(3x + 3h - 1) - (3x - 1)}{h}.\] After doing that, simplify the expression by combining like terms: \[\frac{3h}{h}\]

Step 3: Simplify

Finally, simplify the ratio by dividing \(3h\) by \(h\) to get \(3\).

Key Concepts

Understanding Functions

A function is a special relationship between two sets of numbers or objects, generally expressed as a rule. For each input value (often referred to as "x"), there is exactly one output value (referred to as "f(x)"). This mapping rule is what defines a function. For example, in our given function,
\[ f(x) = 3x - 1 \]we have a linear function since it forms a straight line when graphed. Linear functions have constant rates of change, which means that as you move from one point to another, the change in "y" is proportional to the change in "x".

• The input "x" is multiplied by a constant (in this case 3).
• Then a constant is subtracted or added (here, we subtract 1).

Understanding functions allows you to predict outcomes, graph points, or calculate specific values simply by knowing the "rule" of the function.

Algebra Techniques in Difference Quotients

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is the foundation for understanding the difference quotient, a way to measure the rate at which function outputs change as input values change.
To find the difference quotient, you begin by substituting \(x + h\) into the function. This involves using algebra to expand or manipulate the expression. In the context of our exercise:

• Start by substituting \(x + h\) into the function: \(f(x+h) = 3(x + h) - 1\).
• Use the distributive property (multiplying each term by 3): \(3x + 3h - 1\).

These algebraic steps lead to setting up the difference quotient, an essential tool in calculus for finding the derivative or the rate of change of a function.

Simplifying Expressions

Simplification is the process of reducing an expression to its most basic form while maintaining its original value or meaning. This helps to make the expression easier to work with or understand.
In our problem, after setting up the difference quotient:\[\frac{f(x+h) - f(x)}{h} = \frac{(3x + 3h - 1) - (3x - 1)}{h},\]you'll want to simplify by:

• Combining like terms (in this case, after the subtraction, the terms \(3x\) cancel out).
• Simplifying the resulting expression \(\frac{3h}{h}\) by dividing both numerator and denominator by \(h\).

The simple result \(3\) shows that all the complexity of the initial expression can boil down to a single constant. Simplification not only clarifies a mathematical problem but also enhances understanding and ability to solve similar future problems.