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Chapter 2: Problem 47

Show that if \(f\) and \(g\) are linear functions, then the graphs of \(f \circ g\) and \(g \circ f\) have the same slope.

Short Answer

Expert verified
Given linear functions \(f(x) = m_1x + b_1\) and \(g(x) = m_2x + b_2\), we find the compositions \(f(g(x)) = m_1m_2x + m_1b_2 + b_1\) and \(g(f(x)) = m_1m_2x + m_2b_1 + b_2\). The slopes of both compositions are \(m_1m_2\), so the graphs of \(f \circ g\) and \(g \circ f\) have the same slope.

Step by step solution

01

Write the general form of linear functions

We are given that \(f\) and \(g\) are linear functions. So, we can represent them in the general form of linear functions as follows: \[ f(x) = m_1x + b_1 \] \[ g(x) = m_2x + b_2 \]
02

Find the formula for \(f \circ g\)

To find the formula for the composition \(f(g(x))\), we substitute the equation for \(g(x)\) into the equation for \(f(x)\): \[ f(g(x)) = f(m_2x + b_2) = m_1(m_2x + b_2) + b_1 \] Now, distribute \(m_1\) in the equation above: \[ f(g(x)) = m_1m_2x + m_1b_2 + b_1 \]
03

Find the formula for \(g \circ f\)

Similar to the previous step, we need to substitute the equation for \(f(x)\) into the equation for \(g(x)\) to find the formula for the composition \(g(f(x))\): \[ g(f(x)) = g(m_1x + b_1) = m_2(m_1x + b_1) + b_2 \] Now, distribute \(m_2\) in the equation above: \[ g(f(x)) = m_1m_2x + m_2b_1 + b_2 \]
04

Compare the slopes of \(f \circ g\) and \(g \circ f\)

Upon examining the formulas for \(f(g(x))\) and \(g(f(x))\), we see that the coefficient of \(x\) is the same in both cases. This means that the slope of the graphs of \(f(g(x))\) and \(g(f(x))\) are the same: Slope of \(f(g(x)) = m_1m_2\) Slope of \(g(f(x)) = m_1m_2\) Since the slopes are equal, we have shown that the graphs of \(f \circ g\) and \(g \circ f\) have the same slope.

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