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Chapter 2: Problem 2

In Exercises 1鈥�30, find the domain of each function. $$ f(x)=2(x+5) $$

Short Answer

Expert verified
The domain of the function \(f(x) = 2(x + 5)\) is all real numbers, or in interval notation, \(-\infty, \infty\).

Step by step solution

01

Analyze the function

We see that the function \(f(x) = 2(x + 5)\) is a linear function where x is the variable. Unlike certain functions (such as square root functions or logarithmic functions), a linear function allows for any real number input. There are no limitations on x because any real x value will not make the function undefined. In other words, there are no restrictions on x.
02

Determine the domain

Since there are no restrictions on x, the domain of the function is all real numbers. In interval notation, the domain can be written as \(-\infty, \infty\).

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Key Concepts

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

Linear Function
A linear function is a type of mathematical function that creates a straight line when plotted on a graph. It typically has the form \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. In the function \(f(x) = 2(x + 5)\), it can be expanded to \(f(x) = 2x + 10\), showing it is indeed linear.

Linear functions have some convenient and straightforward properties. These include:
  • Their graphs are straight lines
  • They have constant rate of change (slope)
  • For any x, there is exactly one y
These characteristics mean linear functions are very simple and predictable, which is why they have no restrictions on their domain, allowing for any real number input.
Real Numbers
Real numbers are all the numbers on the number line. They include every kind of number that you can think of, such as whole numbers, fractions, and decimals. Essentially, any number that isn't imaginary (like the square root of a negative number) falls into this category.
With linear functions, like our example \(f(x) = 2(x + 5)\), you can input any real number for \(x\). This is because real numbers cover:
  • Positive numbers (1, 2, 3, ...)
  • Negative numbers (..., -3, -2, -1)
  • Zero (0)
  • Decimals and fractions (0.5, 1/3, etc.)
For our function, using a real number for \(x\) maintains the simplicity and straightforwardness of linear operations, making it a very flexible function with an unrestricted domain.
Interval Notation
Interval notation is a way to describe the set of numbers (or the domain) that a function can accept. For functions where the domain is all real numbers, like our linear function example, interval notation is very descriptive.

The domain of \(f(x) = 2(x+5)\) in interval notation is written as \((-\infty, \infty)\). Here is a quick breakdown of what this means:
  • The \(-\infty\) symbol represents considering all numbers up to negative infinity, which includes all negative real numbers.
  • The \(\infty\) symbol covers all numbers going towards positive infinity, including all positive real numbers.
  • Parentheses \(()\) signify that infinity is not a number that can be actually reached, so it is excluded from real numbers.
This notation efficiently communicates that the function's domain includes every real number without restriction.

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Most popular questions from this chapter

In your own words, describe how to find the distance between two points in the rectangular coordinate system.

Graph \(y_{1}=\sqrt{2-x}, y_{2}=\sqrt{x},\) and \(y_{3}=\sqrt{2-y_{2}}\) in the same \([-4,4,1]\) by \([0,2,1]\) viewing rectangle. If \(y_{1}\) represents \(f\) and \(y_{2}\) represents \(g,\) use the graph of \(y_{3}\) to find the domain of \(f \circ g .\) Then verify your observation algebraically.

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Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.
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