Problem 69 - College Algebra with Modeling and Visualization

Question

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=3 $$

Step-by-Step Solution

Answer

(a) \( f(x+h) = 3 \); (b) Difference quotient is 0.

Step 1: Determine f(x+h)

Given the function \( f(x) = 3 \), it is a constant function, which means its value doesn't change with \( x \). Consequently, \( f(x+h) = 3 \) as well.

Step 2: Construct the difference quotient

The difference quotient is given by the expression \( \frac{f(x+h) - f(x)}{h} \). Substituting the values from our function, we get \( \frac{3 - 3}{h} \).

Step 3: Simplify the difference quotient

Simplify the expression \( \frac{3 - 3}{h} \), which results in \( \frac{0}{h} \). Since the numerator is zero, the simplified difference quotient is 0.

Key Concepts

Understanding Constant Functions

A constant function is a basic type of function where the output value remains unchanged, regardless of the input value. In mathematical terms, this is written as \( f(x) = c \), where \( c \) is a fixed number, also known as the constant.
For example, consider the function \( f(x) = 3 \). This means no matter what value of \( x \) you plug in, the output will always be 3. The graph of a constant function is a horizontal line, which illustrates that the output value remains the same as \( x \) changes.

• The simplicity of constant functions makes them crucial for understanding basic concepts in calculus and algebra.
• They also serve as a benchmark for evaluating more complex functions by providing a stable reference point.

Understanding constant functions is key to grasp dimensions of change in variable functions, as they represent the idea of ‘no change’ or ‘zero variation.’ This is important when working with difference quotients, as the rate of change is zero for constant functions.

Exploring Function Notation

Function notation is a compact and effective way to express equations that denote functions. It typically uses the format \( f(x) \), where \( f \) represents the function and \( x \) is the variable input.
This notation not only tells us that \( f \) is a function but also allows us to understand how an input is transformed into an output. For instance, in the given function \( f(x) = 3 \), \( f(x) \) uses function notation to indicate that whatever value \( x \) takes, the result of the function will be 3.

• It is beneficial for easily communicating complex calculations and transformations.
• The notation allows for easy substitution and manipulation of algebraic expressions.

Using function notation helps simplify the presentation of mathematical relationships and is particularly useful in calculus for derivations and integrations. It clearly distinguishes functions from equations, highlighting the input-output relationship.

Working with Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
In calculus and algebra, expressions play a central role as they represent mathematical objects and operations. For instance, the difference quotient in the exercise \( \frac{f(x+h) - f(x)}{h} \) is a classic example of an algebraic expression.

• They allow mathematicians to generalize operations and create formulas that can be universally applied.
• Simplifying expressions is crucial for problem-solving as it helps in finding solutions that satisfy given conditions.

In the example of the difference quotient with the constant function \( f(x) = 3 \), the simplification of the expression \( \frac{3 - 3}{h} \) demonstrates how algebraic expressions can often be boiled down to simpler forms, aiding in understanding the behavior of a function at a particular point or range. Mastering the manipulation of algebraic expressions is essential for success in mathematics.