/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Sketch a graph of the line. $$... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 5

Sketch a graph of the line. $$f(x)=3$$

Short Answer

Expert verified
The function \(f(x) = 3\) represents a horizontal line intersecting the y-axis at \(y = 3\).

Step by step solution

01

Identify the type of line

Firstly, identify the type of line that the function represents. In this case \(f(x)=3\) represents a horizontal line, as the value of function is constant irrespective of the \(x\) value. This means that for every single value of \(x\), the output of the function is always 3.
02

Draw the graph

To draw the graph, begin by drawing a horizontal line across at \(y=3\) on the y-axis. All points on this line correspond to the equation \(f(x)=3\), since for any given \(x\), the value of \(y\) is 3.

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Key Concepts

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

Understanding Constant Functions
A constant function is one of the simplest types of functions you'll come across in mathematics. It is defined by the equation \(f(x) = c\), where \(c\) is a constant real number. In this type of function, no matter what value \(x\) takes, the function \(f(x)\) will always be equal to the constant \(c\). This is why it is called a "constant" function.

Some characteristics of constant functions include:

  • The output value does not change, regardless of the input \(x\).
  • The graph of a constant function is a horizontal line.
  • Its domain is all real numbers, but the range is just the single value \(c\).
Recognizing a graph as a constant function can significantly simplify your understanding of its behavior. Since the output never changes across its domain, it provides a straightforward visual of a horizontal line.
Linear Equation and its Forms
Linear equations are equations of the first degree, meaning the highest power of the variable \(x\) is 1. The general form of a linear equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

A specific form of a linear equation that is very useful is the slope-intercept form given by \(y = mx + b\). Here:
  • \(m\) represents the slope of the line, indicating the steepness and direction.
  • \(b\) is the y-intercept, showing where the line crosses the y-axis.

For constant functions, the equation takes a simplified linear form without the \(x\) component, such as \(y = 3\). This implies a slope \(m = 0\) since the graph of the equation is horizontal, reflecting no change in \(y\) as \(x\) changes.
Graphing: Plotting a Horizontal Line
Graphing can sometimes seem complex, but it becomes much simpler with constant functions. When graphing a horizontal line like \(f(x)=3\), you are drawing a line on a graph where the y-coordinate remains constant at 3 for all x-coordinates. This means that no matter where you look on the line, \(y\) is always 3.

To plot a horizontal line:
  • Find the value on the y-axis that corresponds to \(y = 3\).
  • Draw a straight line parallel to the x-axis passing through this point.
  • Ensure the line covers the entire graph to show it holds for all \(x\).

Horizontal lines visually demonstrate the concept of a constant function as they have no slope or inclination, maintaining a consistent value throughout.

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Most popular questions from this chapter

Graph the piecewise-defined function using a graphing utility. The display should be in DOT mode. $$f(x)=\left\\{\begin{array}{ll}0.5 x^{2}, & \text { if } x \leq 0 \\\\-x^{2}, & \text { if } x>0\end{array}\right.$$

Applications In this set of exercises you will use the concepts of intersection of lines and linear inequalities to study real-world problems. Cost and Revenue In Exercises \(69-72,\) for each set of cost and revenue functions, (a) find the break-even point and (b) calculate the values of \(q\) for cohich retucnue exceeds cost. $$\begin{aligned} &C(q)=2 q+10\\\ &R(q)=4 q \end{aligned}$$

Graph the function by hand. $$F(x)=\left\\{\begin{array}{ll} 0, & x \leq 1 \\ 2, & x>1 \end{array}\right.$$

Solve the inequality. Express your answer in interval notation. $$\frac{2}{3} x+3 \leq 5$$

Solve the inequality. Express your answer in interval notation. $$0<-x+5<4$$
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