Linear Function Behavior (LFB)

Lines through the origin are peculiar in that they have an expression of the form $f(x) = mx$, where $m$ is a constant (the slope). This formula makes possible the following ``additive" property:

\begin{displaymath}f(x_1 + x_2) = m(x_1 + x_2) = mx_1 + mx_2 = f(x_1) + f(x_2). \end{displaymath}

For the particular choices of $x_1 = x^2$ and $x_2 = 2x$, we would have
\begin{displaymath}f(x^2 + 2x) = f(x^2) + f(2x) = f(x^2) + 2f(x). \end{displaymath}

Despite students' tendancies to treat every function as additive, other functions just do not have this property. Typical mistakes made include:
\begin{displaymath}\sqrt{3x^2 + 2x} = \sqrt{3x^2} + \sqrt{2x} \mbox{\hspace{0.6in} \epsfig{0.5in}{lfb.eps}} \end{displaymath}

\begin{displaymath}(x + 2)^3 = x^3 + 8 \mbox{\hspace{0.6in} \epsfig{0.5in}{lfb.eps}} \end{displaymath}

\begin{displaymath}\ln(2x-1) = \ln(2x) - \ln 1 \mbox{\hspace{0.6in} \epsfig{0.5in}{lfb.eps}} \end{displaymath}

\begin{displaymath}\frac{1}{x+5} = \frac{1}{x} + \frac{1}{5} \mbox{\hspace{0.6in} \epsfig{0.5in}{lfb.eps}} \end{displaymath}


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Thomas L. Scofield 2003-09-04