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Chapter 2: Problem 1

In Exercises 1鈥�30, find the domain of each function. $$ f(x)=3(x-4) $$

Short Answer

Expert verified
The domain of the function \(f(x) = 3(x-4)\) is all real numbers.

Step by step solution

01

Understanding the Function

The function provided is \(f(x) = 3(x-4)\). It is a linear function, and does not contain any denominators, square roots, or logarithms, where we might find some restrictions for the value of x. With this type of function, there are no restrictions for the value of x.
02

Identify the Domain

In the case of a linear function like ours, the function is defined for all real numbers. So, the domain of the function is all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Functions
A linear function is a type of function where the relationship between the input variable, usually denoted as \(x\), and the output, \(f(x)\), forms a straight line when plotted on a coordinate plane.
Linear functions are often expressed in the form \(f(x) = ax + b\) where \(a\) and \(b\) are constants, and \(a\) is not equal to zero.
Characteristics of linear functions include:
  • Constant slope: In a linear function, the rate of change between any two points is constant. This means that if you pick any two points on the line, the change in \(y\) divided by the change in \(x\) (the slope, \(m\)) is the same.
  • Straight-line graph: When you graph a linear function, you will see a straight line. This is because there are no powers of \(x\) other than one.
  • Simplified computations: Calculating values of \(f(x)\) is straightforward because linear formulas do not require dealing with powers greater than one or other complex operations.
Understanding linear functions is fundamental because they are a first step towards grasping more complex functions and mathematical concepts in algebra.
Real Numbers
Real numbers are a major concept in mathematics. They include nearly all the numbers we encounter in everyday life. This set encompasses different types of numbers, each serving particular functions.
Real numbers include:
  • Natural numbers: Positive integers such as 1, 2, 3, etc.
  • Whole numbers: All natural numbers and zero, like 0, 1, 2, 3, etc.
  • Integers: Whole numbers and their negatives, including -3, -2, -1, 0, 1, 2, 3, etc.
  • Rational numbers: Numbers that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).
  • Irrational numbers: Numbers that cannot be expressed as fractions, having non-repeating, non-terminating decimal expansions, such as \(\pi\) and \(\sqrt{2}\).
Real numbers are essential because they are used in measurement, algebra, calculus, and more. Understanding real numbers allows us to grasp the full spectrum of possibilities in various mathematical contexts, particularly in defining the domains of functions.
Function Domains
The domain of a function is the complete set of possible values for the input \(x\) that result in a valid output \(f(x)\).
Determining the domain involves checking for any limitations that might restrict \(x\). These can include:
  • Denominators: If the function includes a fraction, avoid values that make the denominator zero.
  • Square roots: For square root functions, the expression inside the root must be non-negative because we cannot have real square roots of negative numbers.
  • Logarithms: The expression inside a logarithmic function must be greater than zero.
In the context of linear functions like \(f(x) = 3(x-4)\), there are typically no constraints such as division by zero or negative square roots. As a result, the domain is all real numbers (\(-\infty, \infty\)) because you can input any real number into a linear function and obtain a valid output. Knowing how to determine the domain of a function is crucial for graphing and solving functions accurately.

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Most popular questions from this chapter

Find a. \((f \circ g)(x)\) b. the domain of \(f \circ g\) $$f(x)=\frac{x}{x+1}, g(x)=\frac{4}{x}$$

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