Define \({\bf{f}}:\mathbb{R} \to \mathbb{R}\) by \({\bf{f}}\left( {\bf{x}} \right) = {\bf{mx}} + {\bf{b}}\).

1. Show that \({\bf{f}}\) is a linear transformation when \({\bf{b}} = {\bf{0}}\).
2. Find a property of a linear transformation that is violated when \({\bf{b}} \ne {\bf{0}}\).
3. Why is \({\bf{f}}\) called a linear function?

(a)

Given, \(f\left( x \right) = mx + b\)

For the (i) property, let \(x,y \in \mathbb{R}\). Then

\(\begin{aligned} f\left( {x + y} \right) &= m\left( {x + y} \right) + b\\ &= mx + my + b\end{aligned}\)

And

\(\begin{aligned} f\left( x \right) + f\left( y \right) &= mx + b + my + b\\ &= mx + my + 2b\end{aligned}\)

This implies \(f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)\) when \(b = 0\).

For the (ii) property, let \(c,w \in \mathbb{R}\). Then

\(\begin{aligned}{c}f\left( {cw} \right) &= m\left( {cw} \right) + b\\ &= c\left( {mw} \right) + b\end{aligned}\)

And

\(\begin{array}{c}cf\left( x \right) = c\left( {mw + b} \right)\\ = c\left( {mw} \right) + cb\end{array}\)

This implies \(f\left( {cw} \right) = cf\left( w \right)\) when \(b = 0\).

Thus, \(f\) is a linear transformation when \(b = 0\).

(b)

From part (a), \(mx + my + b \ne mx + my + 2b\) when \(b \ne 0\). That is, \(f\left( {x + y} \right) \ne f\left( x \right) + f\left( y \right)\) when \(b \ne 0\).

Also, \(c\left( {mw} \right) + b \ne c\left( {mw} \right) + cb\) when \(b \ne 0\). That is, \(f\left( {cw} \right) \ne cf\left( w \right)\) when \(b \ne 0\).

Hence, both properties \(f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)\) and \(f\left( {cw} \right) = cf\left( w \right)\) are violated when \(b \ne 0\).

(c)

In calculus, we know that a straight line is a linear function. It is a polynomial of degree 1. Therefore, the given function is called a linear function.