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Problem 61 Hard Difficulty

Let $ f $ and $ g $ be linear functions with equations $ f(x) = m_1 x + b_1 $ and and $ g(x) = m_2 x + b_2 $. Is $ f \circ g $ also a linear function? If so, what is the slope of its graph?

Answer

If $f(x)=m_{1} x+b_{1}$ and $g(x)=m_{2} x+b_{2},$ then
\[
(f \circ g)(x)=f(g(x))=f\left(m_{2} x+b_{2}\right)=m_{1}\left(m_{2} x+b_{2}\right)+b_{1}=m_{1} m_{2} x+m_{1} b_{2}+b_{1}.
\]
So $f \circ g$ is a linear function with slope $m_{1} m_{2}$.

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Video Transcript

here we have two linear functions F N g, and we're going to find f of G and see if it's linear or not. So that means we're going to put the G function inside the F function and it will look like this m sub one times m sub two x plus B sub to plus B some to now let's simplify it and see if it looks like it's the equation of a line so we can distribute the EMS of one. And we have m sub one times m sub two times X plus m sub one times be siptu plus B sub to. So what I'm seeing here is I'm seeing some function. Why that has some number times X plus some other number. That, to me, looks like Why equals some slope times X plus some? Why intercept? So, yes, it's linear now because it's linear. We're supposed to say what the slope is. So this right here was the slope, the number that we multiplied by X So the slope is M one times m two