Combining these two results we obtain:

For any three real numbers, a,b,h,

\(f(x)\) is a **linear function** of a real variable \(x\) if it satisfies:

\[f(ax + bh) = af(x) + bf(h). \]

**Exampe Problem: **

Prove that the function \(f(x) = cx\) (for any fixed real constant, \(c\)) is a linear function of \(x\).

**Solution:**

If we multiply \(x\) by any real number \(a\), we obtain:

\[ f(ax) = cax = af(x). \]

If we add a real number \(h\) to \(x\) we obtain:

\[ f(x+h) = cx + ch = f(x) + f(h). \]

**Example Problem:**

If \(f(x)\) and \(g(x)\) are both linear functions of \(x\),

- is \(f(x) + g(x)\) a linear function of \(x?\)
- is \(f(x)g(x)\) a linear function of \(x?\)

**Solution:**

Define \(q(x) = f(x) + g(x)\) and apply the tests of linearity.

\[ q(ax + bh) = f(ax + bh) + g(ax + bh).\]

By the linearity of \(f(x)\) and \(g(x)\) this equals:

\[ af(x) + af(x)+ bf(h) + bg(h) = a(f(x) + g(x)) + b(f(h) + g(h))\]

\[ = aq(x) + bq(h). \]

Therefore \(q(x) = f(x) + g(x)\) is a linear function of \(x\).

Now define \(r(x) = f(x)g(x)\). To keep this simple just multiply \(x\) by \(a\) (a constant). Then,

\[ r(ax) = af(x)ag(x) = a^{2}f(x)g(x). \]

Therefore \(r(x) = f(x)g(x)\) is not a linear function of \(x\).

**Example Problem:**

Is the equation of a straight line a linear function of \(x\)?

A straight line has the equation \(y(x) = mx + b\), where \(m\) is the slope, and \(b\) is the \(y\)-intercept.

**Solution:**

Let's test whether this fulfills the two properties that define a linear function.

First, if we multiply \(x\) by a real number \(c\), then, \(y(c) = mcx + b \neq cy(x)\) unless \(b = 0\).

Next, if we add a real number \(h\), to \(x\) we find:

\[ y(x + h) = mx + mh + b \neq y(x) + y(h)\]

unless \(b = 0\).

So on both counts, the equation of a straight line fails to be a linear equation of \(x\) unless the line passes through the origin.

However, just to make life difficult, the equation of a straight line is often called a **linear equation**. Therefore we must be very careful to distinguish **linear functions** (of \(x\)) from **linear equations** (in \(x\)).

In fact, both linear equations and linear functions play an important role in the theory of polynomials even though, as we will see, most polynomials are neither linear functions nor linear equations.

**Linear Factors:**

We now turn our attention from linear *functions* to linear *equations*.

I.e., equations of the form: \( y = mx +b \).

This equation defines a line that crosses the \(y\)-axis when \(x =0\), and, as long as the slope \(m \neq 0\), crosses the \(x\)-axis when

\[ x = - \frac{b}{m} = \alpha.\]

Hence \(\alpha\) is called the \(x\)-intercept; it is the point at which \(y = 0\).

When \(m \neq 0\) we can express the same line in the form

\[ \frac{y}{m} = \frac{mx + b}{m} = (x - \alpha). \]

Such an expression is called a ** linear factor** of \(x\).

Note, once again, that a linear factor of \(x\) is not, in general, a linear function of \(x\). (It is a linear function only when \(\alpha = 0\).)

**Sample Exercise:**

What is the linear factor of \(x\) associated with the equation:

\[ y = 3x + 6? \]

**Solution:**

\[ \alpha = -\frac{b}{m} = -\frac{6}{3} = -2.\]

Therefore,

\[(x - \alpha) = (x + 2).\]