Function Arithmetic: Using Graphs to Evaluate Various Combinations of Functions at a Given Value
Function Arithmetic Using Graphs
Recall that when you are given an expression such as \((f+g)(2)\), this means “find the outputs of \(f\) and \(g\) on input \(2\), then add the outputs.” More succinctly, \((f+g)(2)=f(2)+g(2)\). This interpretation, along with how to evaluate the expression, works exactly the same way regardless of how your functions are represented (i.e. its the same if your functions are defined by a graph, table, set, rule, machine, etc.)
Note that \((f+g)(1)=f(1)+g(1)\) so we need to find \(f(1)\) and \(g(1)\) first. \(f\) is the green wiggly function, and the \(y\)-value of \(f\) at \(x=1\) is about \(1.25\). Similarly, the \(y\)-value of the blue function \(g\) at \(x=1\) is \(2.75\)ish. So, we can estimate the value of \((f+g)(1)\) as
$$\begin{align}(f+g)(1)&=f(1)+g(1)\\&=1.25+2.75\\&=4\end{align}$$
Similarly, to compute \((f/g)(2)=\frac{f(2)}{g(2)}\) we first need to compute \(f(2)\) and \(g(2)\) using the graph. Note that at \(x=2\) the green function \(f\) has \(y\)-value roughly \(0.5)\) and the blue function \(g\) has \(y\)-value roughly equal to \(1.25\). So, \((f/g)(2)\) is therefore
$$\begin{align}\left(\frac{f}{g}\right)(2)&=\frac{f(2)}{g(2)}\\&=\frac{0.5}{1.25}\\&=0.4\end{align}$$
Overall, performing function arithmetic using graphs (or any other means, really) involves nothing more than computing each function’s \(y\)-value at the given \(x\) value, and then combining the \(y\)-values as indicated by the operation, such as \(+\), \(x\), \(/\), etc.
