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Linear Functions and Linear Equations

08 March 2014 By Gideon In Polynomials
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 A linear function, \(f(x)\), of a real number, \(x\), is defined by two properties:

  • for any real number, \(a\), \(f(ax) = a(f(x));\)
  • for any real number, \(h\), \(f(x+h) = f(x) + f(h).\)

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,

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

 

Read 1747 times Last modified on Sunday, 09 March 2014 16:49

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