Question

What do all members of the family of linear functions\({\bf{f}}\left( {\bf{x}} \right){\bf{ = 1 + m}}\left( {{\bf{x + 3}}} \right)\)have in common? Sketch several members of the family.

Short answer

All linear functions have a common point\(\left( { - 3,1} \right)\).

Step-by-step solution

Given data

\(f\left( x \right) = 1 + m\left( {x + 3} \right)\)

Solution

It is clear that\(m\)is arbitrary in\(f\left( x \right)\).

For a different value of\(m\)we get different linear functions.

Draw the graph of the function with different\(m\)values.

Below graph is drawn for\(m = 1,2, - 1, - 2\).

All lines intersect at\(\left( { - 3,1} \right)\).

Hence, all linear functions have a common point\(\left( { - 3,1} \right)\).

The graph is shown below

The graph is drawn as: