Question

Let f and g be the linear functions with equations \(f\left( x \right) = {m_{\bf{1}}}x + {b_{\bf{1}}}\) and \(g\left( x \right) = {m_{\bf{2}}}x + {b_{\bf{2}}}\). Is \(f \circ g\) also a linear function? If so, what is the slope of its graph?

Short answer

Yes, \(f \circ g\) is a lienar function and slope is \({m_1}{m_2}\).

Step-by-step solution

Find the composite function \(f \circ g\)

Thecomposite function\(f \circ g\) can be expressed as:

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

Check for the properties of \(f \circ g\)

So, the function \(f \circ g\left( x \right) = {m_1}{m_2}x + {m_1}{b_2} + {b_1}\) is a linear function and the slope of the graph of the function is \({m_1}{m_2}\).

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