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Chapter 2: Problem 48

Show that a linear function is increasing if and only if the slope of its graph is positive.

Short Answer

Expert verified
A linear function is represented by the equation \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. A function is increasing if, for any two points \(x_1\) and \(x_2\), if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). We can prove that a linear function is increasing if and only if the slope is positive by showing that a positive slope implies an increasing function, and vice versa. For an increasing function, we can write the inequality \(mx_1 + b < mx_2 + b\) which results in a positive slope value, and for a positive slope, we can derive that \(f(x_2) - f(x_1) > 0\), proving an increasing function.

Step by step solution

01

Define a Linear Function

A linear function is an equation of the form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. The slope 'm' indicates the steepness of the line and the direction in which the line moves alongside the x-axis. The y-intercept 'b' gives the point where the line crosses the y-axis.
02

Define an Increasing Function

A function is said to be increasing if, for any two points \(x_1\) and \(x_2\) in the domain, if \(x_1 < x_2\), then \(f(x_1) < f(x_2)\). In other words, the output values of the function increase as the input values increase.
03

Proof: A Positive Slope Implies an Increasing Linear Function

Let's assume the slope 'm' of the linear function is positive, i.e., \(m > 0\). For any two values in the domain, say \(x_1\) and \(x_2\), and \(x_1 < x_2\), We want to show that \(f(x_1) < f(x_2)\). Using the linear function equation, we have: \(f(x_1) = mx_1 + b\) \(f(x_2) = mx_2 + b\) Now, subtract the first equation from the second: \(f(x_2) - f(x_1) = m(x_2 - x_1)\) Since \(x_2 > x_1\), it means that \((x_2 - x_1) > 0\). We already know that our slope \(m > 0\). Therefore, the product of two positive numbers is also positive: \(f(x_2) - f(x_1) = m(x_2 - x_1) > 0\) Thus, we can conclude that if the slope is positive, the function is increasing.
04

Proof: An Increasing Linear Function Implies a Positive Slope

Conversely, let's assume that the linear function is increasing. For any two values in the domain, say \(x_1\) and \(x_2\), and \(x_1 < x_2\), we have \(f(x_1) < f(x_2)\). Using the linear function equation, we have: \(f(x_1) = mx_1 + b\) \(f(x_2) = mx_2 + b\) Now we can write this inequality: \(mx_1 + b < mx_2 + b\) Subtract 'b' from both sides and divide by \((x_2 - x_1)\), which is positive as \(x_2 > x_1\). We have: \(m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} > 0\) Thus, we can conclude that if the function is increasing, the slope is positive. Therefore, a linear function is increasing if and only if the slope of its graph is positive.

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Most popular questions from this chapter

$$ \text { Suppose } p(x)=2 x^{6}+3 x^{5}+5 $$ (a) Show that if \(\frac{M}{N}\) is a zero of \(p\), then $$ 2 M^{6}+3 M^{5} N+5 N^{6}=0 $$ (b) Show that if \(M\) and \(N\) are integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\), then \(5 / M\) and \(2 / N\) are integers. (c) Show that the only possible rational zeros of \(p\) $$ \text { are }-5,-1,-\frac{1}{2}, \text { and }-\frac{5}{2} \text { . } $$ (d) Show that no rational number is a zero of \(p\).

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