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Chapter 2: Problem 12

A function \(f\) is given. (a) Sketch a graph of \(f .\) (b) Use the graph to find the domain and range of \(f\). $$f(x)=3 x-2$$

Short Answer

Expert verified
The domain and range of the function \( f(x) = 3x - 2 \) are both \( (-\infty, \infty) \).

Step by step solution

01

Understand the Function Type

The given function is \( f(x) = 3x - 2 \). This is a linear function because it is in the form of \( f(x) = mx + b \) where \( m = 3 \) (slope) and \( b = -2 \) (y-intercept).
02

Sketch the Graph

To plot the graph of \( f(x) = 3x - 2 \), begin by identifying the y-intercept, which is the point where \( x = 0 \). Here, \( f(0) = 3(0) - 2 = -2 \), so the y-intercept is (0, -2). Next, using the slope \( m = 3 \), move up 3 units and 1 unit to the right starting from the y-intercept to plot another point. Therefore, from (0, -2), the next point is (1, 1). Draw a straight line through these points extending in both directions.
03

Determine the Domain

The domain of a linear function like \( f(x) = 3x - 2 \) is all real numbers because you can input any real number into the function. Therefore, the domain is \( (-\infty, \infty) \).
04

Determine the Range

The range of a linear function is also all real numbers since the value of \( f(x) \) can take any real number depending on the value of \( x \). Therefore, the range is \( (-\infty, \infty) \).
05

Conclusion

From the graph of \( f(x) = 3x - 2 \), we confirm that it is a straight line. Thus, the domain covers all x-values from negative to positive infinity, and the range covers all y-values, also from negative to positive infinity.

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Key Concepts

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

Function Graphs
A function graph visually represents the relationship between the input and output of a function. For a linear function like \( f(x) = 3x - 2 \), its graph is a straight line. This happens because linear functions have a constant rate of change or a constant slope.
To sketch the graph, we need two main things: the slope and the y-intercept. The slope, in our function, is 3, which shows the line rises 3 units upwards for each 1 unit it goes to the right. The y-intercept is -2, which means the line crosses the y-axis at (0, -2).
Here is how you would plot this.
  • Start at the y-intercept (0, -2).
  • From (0, -2), making use of the slope, move 3 units up and 1 unit to the right to locate another point on the line, which would be (1, 1).
  • Draw a straight line through these two points to extend in both directions infinitely.
This line represents the graph of the function.
Domain and Range
Understanding the domain and range of a function is crucial as it informs us about what values the function can handle and what outputs it can produce.
For linear functions like \( f(x) = 3x - 2 \), the domain is the set of all possible x-values. Since you can input any real number into a linear equation without restrictions, the domain of this function is all real numbers. This is represented as \( (-\infty, \infty) \).
The range, on the other hand, refers to the set of possible y-values. For a linear function that extends infinitely in both vertical directions, this encompasses all real numbers as well. Therefore, just like the domain, the range is \( (-\infty, \infty) \).
This means for every real number input into \( f(x) \), there is a corresponding real number output, depicted by the graph covering every y-value.
Graphing Techniques
Graphing techniques are often used to accurately represent mathematical functions visually. Here's a simple guide.
When graphing linear functions such as \( f(x) = 3x - 2 \), follow these steps:
  • Identify the y-intercept from the function equation. In our case, it's -2.
  • Plot the y-intercept on the graph at (0, -2).
  • Use the slope to find another point. The slope tells you to move up 3 and right 1 from the y-intercept to find the next point, (1, 1).
  • Draw a line through these two points. Ensure it extends across the entire graph, as it represents all possible values of x and y.
When you consistently apply these graphing techniques, you can easily understand and depict linear functions in detail.

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Most popular questions from this chapter

When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature \(T\) of the soup is a function of time \(t .\) The table below gives the temperature (in "F) of a bowl of soup \(t\) minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly? $$\begin{array}{|c|c||c|c|} \hline t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) & t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) \\ \hline 0 & 200 & 35 & 94 \\ 5 & 172 & 40 & 89 \\ 10 & 150 & 50 & 81 \\ 15 & 133 & 60 & 77 \\ 20 & 119 & 90 & 72 \\ 25 & 108 & 120 & 70 \\ 30 & 100 & 150 & 70 \\ \hline \end{array}$$

DISCUSS DISCOVER: Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $$g(x)=\sqrt{f(x)}$$ where \(f(x) \geq 0\) for all \(x .\) Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

You have a \(\$ 50\) coupon from the manufacturer that is good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a \(20 \%\) discount on all cell phones. Let \(x\) represent the regular price of the cell phone. (a) Suppose only the \(20 \%\) discount applies. Find a function \(f\) that models the purchase price of the cell phone as a function of the regular price \(x\). (b) Suppose only the \(\$ 50\) coupon applies. Find a function \(g\) that models the purchase price of the cell phone as a function of the sticker price \(x\). (c) If you can use the coupon and the discount, then the purchase price is either \((f \circ g)(x)\) or \((g \circ f)(x),\) depending on the order in which they are applied to the price. Find both \((f \circ g)(x)\) and \((g \circ f)(x) .\) Which composition gives the lower price?

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if }|x| \leq 1 \\ 1 & \text { if }|x|>1 \end{array}\right.$$

Graphing Functions Sketch a graph of the function by first making a table of values. $$r(x)=3 x^{4}$$
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