Problem 5 - Basic Technical Mathematics with Calculus

Question

Find the domain and range of the given functions. In Exercises 11 and \(12,\) explain your answers. $$f(x)=x+5$$

Step-by-Step Solution

Answer

The domain and range of \( f(x) = x + 5 \) are both \( \mathbb{R} \), all real numbers.

Step 1: Determine the Domain of f(x)

The domain of a function refers to all possible input values (x-values) that the function can accept. For a linear function like \( f(x) = x + 5 \), there are no restrictions on the values of \( x \). This means \( x \) can be any real number. Therefore, the domain of \( f(x) \) is \( \mathbb{R} \), which represents all real numbers.

Step 2: Determine the Range of f(x)

The range of a function is all possible output values (f(x) values) that the function can produce. Since \( f(x) = x + 5 \) is a linear function with a non-zero slope (slope = 1), the function produces all real numbers as outputs as \( x \) varies over all real numbers. Therefore, the range of \( f(x) \) is also \( \mathbb{R} \), the set of all real numbers.

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions you will encounter. It has a specific form, usually written as \[ f(x) = mx + b \] where:

• \( m \) represents the slope of the line.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

Linear functions result in straight lines when graphed, making them easier to analyze. Their characteristics make them foundational for understanding more complex mathematical concepts. In the example of the function \( f(x) = x + 5 \), the slope \( m \) is 1, and the y-intercept \( b \) is 5. This line will perfectly increase or decrease in equal steps, reflecting the simplicity and predictability of linear functions.

Real Numbers

Real numbers encompass a broad category of numbers, including rational and irrational numbers. They are crucial for defining domains and ranges in functions. Real numbers can be positive, negative, or zero. They include integers, fractions, and non-terminating, non-repeating decimals.

• Rational numbers are those numbers which can be expressed as a fraction of two integers, like 1/2 or -3.
• Irrational numbers cannot be represented as simple fractions. Examples include \( \pi \) and the square root of 2.

When discussing linear functions, both the domain and range often encompass all real numbers because linear functions extend infinitely in both directions on the graph. Thus, each real number input in the domain maps to a real number output in the range. For example, in \( f(x) = x + 5 \), both the domain and range are \( \mathbb{R} \), the set of all real numbers.

Function Analysis

Function analysis is a process used to gain insights about functions, including understanding their behavior, domain, and range. Analyzing functions helps to predict how changes in input affect the output, which is essential in mathematical problem solving.

• The domain represents all possible input values (x) for a function. For linear functions like \( f(x) = x + 5 \), the domain is all real numbers because there are no restrictions on what x can be. This means every x-value you substitute into the function will give you a legitimate answer.
• The range indicates all possible output values (f(x)) the function can generate. For linear functions, since they extend infinitely in both upward and downward directions, the range is also all real numbers.

Function analysis ensures students predictably convert the input values into output outcomes, giving a comprehensive understanding of how mathematical functions behave. This understanding aids in tackling real-world problems where mathematical modeling is required.