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Chapter 5: Problem 55

Graph each function. Give the domain and range. \(f(x)=-2 x+1\)

Short Answer

Expert verified
The domain is \((-鈭�, 鈭�)\) and the range is \((-鈭�, 鈭�)\).

Step by step solution

01

Understand the function

Recognize that the function given is a linear function of the form \(f(x) = mx + b\). Here, \(m = -2\) and \(b = 1\).
02

Identify the slope and y-intercept

The coefficient of \(x\) is \(-2\), which is the slope of the line. The \(y\)-intercept is \(1\) (where the line crosses the \(y\)-axis).
03

Plot the y-intercept

Begin by plotting the point \((0, 1)\) on the graph. This is the point where the line crosses the \(y\)-axis.
04

Use the slope to find another point

From the point \((0, 1)\), use the slope \(-2\). This means for every 1 unit increase in \(x\), \(f(x)\) decreases by 2 units. Hence, from \((0, 1)\), moving 1 unit to the right (\(x = 1\)), plot the point \((1, -1)\).
05

Draw the line

Draw a straight line through the points \((0, 1)\) and \((1, -1)\) to extend in both directions.
06

Determine the domain

Since the function is linear, the domain is all real numbers. This can be written as \((-\b+infty}, \b+infty})\).
07

Determine the range

Similarly, the range of a linear function is all real numbers. This can be written as \(\b+(-\b+infty}, \b+infty})\).

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Key Concepts

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

Domain and Range
The domain and range of a linear function are core concepts in understanding how the function behaves.
For a linear function like \(f(x) = -2x + 1\), both the domain and range are all real numbers. This is because a linear function extends infinitely in both the positive and negative directions of the x-axis and y-axis.
Simply put, there are no restrictions on the values that x and y can take.
You can write the domain as \((-\infty, \infty)\)\. Similarly, the range is also \((-\infty, \infty)\).
This makes it easy to understand that linear functions cover all possible input (x) and output (y) values.
Y-Intercept
The y-intercept of a linear function is the point where the graph crosses the y-axis.
In our function \(f(x) = -2x + 1\), the y-intercept is 1, because this is the constant term that stands alone without an x attached.
To graphically find this, plot the point \(0, 1\) on the graph.
This means when x is zero, f(x) is equal to 1.
Understanding the y-intercept helps in quickly plotting the initial point of the linear graph.
Slope
The slope of a linear function indicates how steep the line is and the direction it moves.
For our function \(f(x) = -2x + 1\), the slope is -2. The slope is the coefficient of x in the equation.
This tells us that for every 1 unit increase in x, the value of f(x) decreases by 2 units.
Starting from the y-intercept \(0, 1\), if you move 1 unit to the right (x = 1), f(x) drops to -1.
This can be observed by plotting the point (1, -1).
Connecting the two points \(0, 1\) and \(1, -1\) with a straight line will give you the graph of the function.

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Most popular questions from this chapter

Solve each problem. The U.S. budget first passed \(\$ 1,000,000,000\) in \(1917 .\) Seventy years later, in \(1987,\) it exceeded \(\$ 1,000,000,000,000\) for the first time. The budget request for fiscal-year 2009 was \(\$ 3,100,000,000,000 .\) If stacked in dollar bills, this amount would stretch \(210,385 \mathrm{mi},\) almost \(90 \%\) of the distance to the moon. Write the four boldfaced numbers in scientific notation.

Predict the result the calculator will give for each screen. (Use the usual scientific notation to write your answers.) $$ (1.5 E 12) *(5 E-3) $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{(-8 x y) y^{3}}{4 x^{5} y^{4}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ 7 k^{2}(-2 k)\left(4 k^{-5}\right)^{0} $$

Solve each problem. In \(2009,\) the national debt of the U.S. government was about \(\$ 11.5\) trillion. Using 307.0 million as the population for that year, about how much was this per American? Write the amount in standard notation to the nearest dollar.
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