Functions of One Variable

You can use the same procedure to define a function .

$\blacktriangleright$ To define the function f whose value at x is ax2 + bx + c

1.
Write f (x) = ax2 + bx + c.

2.
Place the insertion point in the equation.

3.
Click the New Definition button on the Compute toolbar, or from the Define submenu, choose New Definition.

Now the symbol f represents the defined function and behaves like a function. Making the definition f (y) = ay2 + by + cdefines the same function. The symbol used for the function argument in making the definition does not matter. To see what happens, define a function f using the equation f (x) = ax2 + bx + c, and apply Evaluate to f (t) to get f (t) = at2 + bt + c.

Note    Making the definition f (y) = ay2 + by + cdefines the same function as the definition f (x) = ax2 + bx + c. The symbol used for the function argument in making the definition does not matter. This point illustrates the subtle but essential difference between expressions and functions. In particular, the two expressions y = x2 + $\sqrt{{x}}$ and y = t2 + $\sqrt{{t}}$ are different (because y gets replaced by an expression in x under the first definition and y gets replaced by an expression in t under the second definition). However, the functions f (x) = x2 + $\sqrt{{x}}$ and f (t) = t2 + $\sqrt{{t}}$ are identical.

If g and h are previously defined functions, then the following equations are examples of legitimate definitions:

f (x) = 2g(x)  
f (x) = g(x) + h(x)  
f (x) = g(x)h(x)  
f (x) = g(h(x))  
   


 

Make a definition for g and h, and then apply Evaluate to f (t) for each definition of f. Each time you redefine f, the new definition replaces the old one. Also, once you have defined both g(x) and f (x) = 2g(x), then changing the definition of g(x) redefines f (x).

Note    The algebra of functions includes objects such as f±g, fog, fg, and f-1. For the value of f + g at x, write f (x) + g(x); for the value of the composition of two defined functions f and g, write f (g(x)) or $\left(\vphantom{ f\circ g}\right.$fog$\left.\vphantom{ f\circ g}\right)$$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$; and for the value of the product of two defined functions, write f (x)g(x). You can obtain the inverse for some functions f (x) by applying Solve + Exact to the equation f (y) = x and specifying y as the Variable to Solve for.


\begin{example}
Define $f(x)=x^{2}+3x+5$\ and $g(x)=x^{3}-1$. Then, \textsf{Eva...
...) \right) \right) &=&7495808-6124745i
\end{eqnarray*}\par
\strut
\end{example}


\begin{example}
You can sometimes find the inverse of a
\index{Inverse@Inverse...
...the function $g$\ is the inverse of the function $f$.
\par
\strut
\end{example}


\begin{example}
With Maple, you can find several values at once by
applying a f...
...
4 & 4163
\end{array}
\right) \vspace{6pt}
\end{displaymath}
\end{example}

Tip    To make a table that will print with lines, create a table with Insert + Table    (or click itbpF27.5pt24.8125pt7pttable.wmf) and copy the information into the table by selecting, clicking and dragging each piece of data. Choose Edit + Properties and add lines according to instructions in the tabbed Table Properties dialog. This is only for purposes of editing—a table does not behave mathematically as a matrix.