Problem 38 - Calculus with Concepts in Calculus

Question

Determine the range of the function. $$ f(x)=3 x-2 $$

Step-by-Step Solution

Answer

The range is all real numbers, \( (-\infty, \infty) \).

Step 1: Identify the Function Type

The given function is a linear function of the form \( f(x) = mx + b \), where \( m = 3 \) and \( b = -2 \). Linear functions create a straight line when graphed.

Step 2: Understand the Range of Linear Functions

For any linear function \( f(x) = mx + b \), the output (or range) is not restricted, meaning it can take any real number value. This is because a line extends infinitely in both directions on a graph.

Step 3: Determine the Range

Since the linear function extends infinitely and is not restricted by a domain or transformation, the range of the function \( f(x) = 3x - 2 \) is all real numbers.

Key Concepts

Linear Functions

A linear function is one of the most fundamental types of functions in mathematics. It is characterized by a constant rate of change, which produces a straight line when graphed. The general form of a linear function is given by: \[ f(x) = mx + b \] Where:

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

A key point of linear functions is that they represent direct proportionality between variables. As one variable changes, the other changes in a consistent manner. For example, increasing \( x \) by 1 unit will consistently increase \( y \) by the same amount determined by the slope \( m \). Moreover, because they do not involve powers or products of variables, their graphical representations are always straight lines.

Function Type

Understanding the type of function you are dealing with is crucial for solving many mathematical problems. Functions can be divided into several types based on their structure and characteristics. Linear functions, such as \( f(x) = 3x - 2 \), are one such type. Some other common function types include:

• Quadratic functions, which form parabolas and have the general form \( ax^2 + bx + c \).
• Polynomial functions, which can have multiple degrees and include terms like \( ax^n \).
• Exponential functions, where the variable is an exponent, such as \( a^x \).
• Trigonometric functions that relate to the angles of triangles.

Recognizing the function type helps to understand its behavior, such as how it changes (derivative), plot (graph shape), and solve (find roots). This foundational understanding makes it easier to work through problems and identify possible solutions.

Real Numbers

In mathematics, real numbers include an extensive range of values, such as integers, fractions, and irrational numbers. Essentially, real numbers encompass any number that can be found on a continuous line, like the number line you see in algebra classes. They are denoted by the symbol \( \mathbb{R} \), indicating their comprehensive nature, covering:

• Natural numbers like 1, 2, 3, and so forth.
• Whole numbers, which include natural numbers and zero.
• Integers that encompass both positive and negative numbers, and zero.
• Rational numbers, which can be expressed as a fraction of two integers.
• Irrational numbers such as \( \pi \) and \( \sqrt{2} \) that cannot be written as simple fractions.

When considering the range of a linear function like \( f(x) = 3x - 2 \), it is important to note that the output can be any real number. This unlimited range is because the line does not stop, extending indefinitely across the number line. Therefore, no real numbers are off-limits when representing the range of linear functions.