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Chapter 12: Problem 59

What is the domain of a linear function?

Short Answer

Expert verified
The domain of a linear function is all real numbers, denoted as \((-\infty, +\infty)\) or \(\{x \in \mathbb{R}\}\), as there are no restrictions on the input values for a linear function.

Step by step solution

01

Definition of a Linear Function

A linear function is a function of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. It represents a straight line on the Cartesian plane when it is graphed. The slope \(m\) determines the direction and steepness of the line, while the y-intercept \(b\) determines the point where the line crosses the y-axis.
02

Domain of a Function

The domain of a function is the set of all possible input values, or in other words, the set of all x-values for which the function is defined.
03

Domain of a Linear Function

In the case of a linear function, the function is defined for all possible x-values. There are no restrictions or limitations on the input values for a linear function. The graph of a linear function extends indefinitely in both directions along the x-axis. Therefore, the domain of a linear function is all real numbers, which can be denoted as \((-\infty, +\infty)\) or using the set notation \(\{x \in \mathbb{R}\}\).

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Most popular questions from this chapter

\(R(x)=80 x\) is the revenue function for the sale of \(x\) bicycles, in dollars. The cost to manufacture \(x\) bikes, in dollars, is \(C(x)=60 x+7000\) (IMAGE CANT COPY) a) Find the profit function, \(P(x),\) that describes the manufacturer's profit from the sale of \(x\) bicycles. b) What is the profit from the sale of 500 bicycles?

For each pair of functions, find a) \(\left(\frac{f}{g}\right)(x)\) and b \(\left(\frac{f}{g}\right)(-2)\) Identify any values that are not in the domain of \(\left(\frac{f}{g}\right)(x)\). $$f(x)=x^{2}-5 x-24, g(x)=x-8$$

Graph the following greatest integer functions. $$h(x)=[x-1]$$

Graph the following piecewise functions. $$f(x)=\left\\{\begin{array}{cc}2 x-4, & x>1 \\\\-\frac{1}{3} x-\frac{5}{3}, & x \leq 1\end{array}\right.$$

Determine the domain of each function. $$r(t)=t^{3}-7 t^{2}+t+4$$
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