# Linear Functions

A linear function is a function of the form

\[f\left( x \right) = ax + b,\,\,\,a \ne 0\]

If *a* is 0, then we will think of *f* as a constant rather than as a linear function.

The domain of a linear function is the set of all real numbers, and so its range:

\[D =\mathbb{R} ,\,\,\,R =\mathbb{R} \]

The following figure shows \(f\left( x \right) = 2x+ 3\) and \(g\left( x \right) = 4 - x\) plotted on the same axes:

Note that both functions take on real values for all values of *x*, which means that the domain of each function is the set of all real numbers. Look along the *x*-axis to confirm this. For every value of *x*, we have a point on the graph.

Also,the output for each function ranges continuously from negative infinity to positive infinity, which means that the range of either function is also \(\mathbb{R}\) . This can be confirmed by looking along the *y*-axis, which clearly shows that there is a point on each graph for every *y*-value.

As we have seen in coordinate geometry, the slope of the line in the graph of \(f\left( x \right) = ax + b\) will be \(m = a\), and it will pass through the point \(\left( {0,b} \right)\), because its *y*-intercept will be *b*.

**Example 1:** Plot the graph of

\[f\left( x \right) = \left\{ \begin{array}{l}x + 2, & x \in \left[ { - 2,1}\right)\\2x - 3, & x \in \left[ {1,2}\right]\end{array} \right.\]

**Solution:**This piecewise function is linear in both the indicated parts of its domain.The graph is shown below: