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Chapter 1: Problem 39

If \(f(x)\) is a linear function, \(f(1)=0,\) and \(f(2)=1,\) what is \(f(3) ?\)

Short Answer

Expert verified
f(3) = 2

Step by step solution

01

Identify the linear function

A linear function can be written in the form: f(x) = mx + b, where m is the slope and b is the y-intercept.
02

Set up the equations using given points

Given the points (1, 0) and (2, 1), substitute them into the linear function form: f(1) = m(1) + b = 0 f(2) = m(2) + b = 1
03

Solve for the slope (m)

Subtract the first equation from the second equation to eliminate b and solve for m: (m * 2 + b) - (m * 1 + b) = 1 - 0 m = 1
04

Solve for the y-intercept (b)

Substitute m back into one of the original equations to solve for b: m * 1 + b = 0 1 * 1 + b = 0 b = -1
05

Write the equation of the linear function

Now that m and b are found, write the linear function: f(x) = x - 1
06

Find f(3)

Substitute x = 3 into the linear function: f(3) = 3 - 1 f(3) = 2

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Key Concepts

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

slope
In a linear function, the 'slope' (m) tells us how steep the line is. It describes how much the y-value (vertical change) changes for every one unit increase in the x-value (horizontal change). When given two points, \(x_1, y_1\) and \(x_2, y_2\), we can determine the slope using the formula: \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). By understanding the slope, you can predict how the line behaves and visualize its direction.
y-intercept
The 'y-intercept' (b) of a linear function is the point where the line crosses the y-axis. This occurs when the x-value is zero. So, the linear function equation \( f(x) = mx + b \) simplifies to \( y = b \) when \( x = 0 \). To find the y-intercept, substitute the slope and one point into the linear equation and solve for b. The y-intercept tells you the starting point of the line on the y-axis.
equation of a line
The equation of a linear function is typically written in the form \( y = mx + b \), where m is the slope and b is the y-intercept. This equation represents a straight line on a graph. For example, given points (1, 0) and (2, 1), the process involves:
  • Finding the slope \( m \)
  • Finding the y-intercept \( b \)
  • Writing the final equation
In this case, the equation becomes \( f(x) = x - 1 \). With this equation, you can find the y-value of any x-value in the function's domain.
solving equations
Solving equations is a fundamental skill in algebra. It involves finding the value of variables that make the equation true. For linear functions, you often solve for the slope (m) and y-intercept (b) first:
  • Set up the equations using given points
  • Solve for the slope by eliminating the y-intercept
  • Substitute the slope back to find the y-intercept
By solving these, you obtain the linear function's equation. After that, you can easily find any value of the function by substituting x-values, just like finding \( f(3) \) in this exercise, where \( f(3) = 2 \).

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Most popular questions from this chapter

Let \(y\) denote the percentage of the world population that is urban \(x\) years after 2014. According to data from the United Nations, 54 percent of the world's population was urban in 2014 , and projections show that this percentage will increase to 66 percent by 2050. Assume that \(y\) is a linear function of \(x\) since 2014. (a) Determine \(y\) as a function of \(x.\) (b) Interpret the slope as a rate of change. (c) Find the percentage of the world's population that is urban in 2020. (d) Determine the year in which \(72 \%\) of the world's population will be urban.

Compute the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ Simplify your answer as much as possible. $$f(x)=-2 x^{2}+x+3$$

Let \(y\) denote the average amount claimed for itemized deductions on a tax return reporting \(x\) dollars of income. According to Internal Revenue Service data, \(y\) is a linear function of \(x .\) Moreover, in a recent year income tax returns reporting \(\$ 20,000\) of income averaged \(\$ 729\) in itemized deductions, while returns reporting \(\$ 50,000\) averaged \(\$ 1380 .\) (a) Determine \(y\) as a function of \(x.\) (b) Graph this function in the window \([0,75000]\) by \([0,2000].\) (c) Give an interpretation of the slope in applied terms. (d) Determine graphically the average amount of itemized deductions on a return reporting \(\$ 75,000.\) (e) Determine graphically the income level at which the average itemized deductions are \(\$ 1600 .\) (f) If the income level increases by \(\$ 15,000,\) by how much do the average itemized deductions increase?

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x.\) $$f(x)=\frac{x^{2}+x-12}{x+4}, x \neq-4$$
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