Question

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

Short answer

\(\left( {f \circ g} \right)\)is a linear function with slope \({m_1}{m_2}\).

Step-by-step solution

Introduction

Two functions can be expressed as a combination of each other with the addition, subtraction, multiplication, and divide operation. One or more operations can be applied at a time.

If f and g are two functions they can be written as,

\(\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right),\left(f\right)\left(x\right),\left({f}/{g}\;\right)\left(x\right)\)

Given

\(\begin{array}{l}f(x) = {m_1}x + {b_1} \\g(x) = {m_2}x + {b_2}\end{array}\)

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Slope of the graph

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\(\begin{array}{l}\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\\\left( {f \circ g} \right)\left( x \right) = f\left( {{m_2}x + {b_2}} \right)\\\left( {f \circ g} \right)\left( x \right) = {m_1}\left( {{m_2}x + {b_2}} \right) + {b_1}\\\left( {f \circ g} \right)\left( x \right) = {m_1}{m_2}x + {m_1}{b_2} + {b_1}\end{array}\)

So,\(\left( {f \circ g} \right)\)is a linear function with slope \({m_1}{m_2}\). \(\)

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