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Section 9.4 Operations on Functions
Once one or more functions are assigned, we can use commands on those functions for various options, such as plotting them.
\begin{equation*}
\displaystyle f\, := \,x\mapsto 2\,{x}^{3}
\end{equation*}
Multiple functions can be combined through composition to create new expressions.
\begin{equation*}
\displaystyle g\, := \,t\mapsto t+1
\end{equation*}
\begin{equation*}
\displaystyle 2\, \left( t+1 \right) ^{3}
\end{equation*}
\begin{equation*}
\displaystyle 2\,{t}^{3}+6\,{t}^{2}+6\,t+2
\end{equation*}
Example 9.1 . Average Rate of Change of a Function over an Interval.
In this example, we will find the average rate of change of the function
\(f(x) = -2x^3 + 25x^2 + 15\) over the interval
\([2,7]\text{.}\) We begin by defining the function:
> f(x) := -2*x^3 + 25*x^2 + 15;
\begin{equation*}
\displaystyle f\, := \,x\mapsto -2\,{x}^{3}+25\,{x}^{2}+15
\end{equation*}
Once the function is defined, we can calculate the average rate of change over an interval \([a,b]\) by using the formula
\begin{equation*}
\frac{f(b)-f(a)}{b-a}\text{.}
\end{equation*}
In this case, we let \(a=2\) and \(b=7\text{:}\)
\begin{equation*}
\displaystyle 91
\end{equation*}
The average rate of change over the interval
\([2,7]\) is
\(91\text{.}\) The units of this rate would be given in (units of
\(y\) )/(units of
\(x\) ).
Example 9.2 . Plotting Transformations of Functions.
> g(x) := 2*sin(Pi*x); plot( g(x), x=0..4, y=-4..4 );
> plot( [g(x), 0.5*g(x)], x=0..4, y=-4..4);
> plot( [ g(x), abs(g(x)) ], x=0..4, y=-4..4);
