3.5 Transformation of functions  (Page 8/21)

Symbolically, we could write

See [link] for a graphical comparison of the original population and the compressed population.

The formula $g(x)=f\left(\frac{1}{2}x\right)$ tells us that the output values for $g$ are the same as the output values for the function $f$ at an input half the size. Notice that we do not have enough information to determine $g(2)$ because $g(2)=f\left(\frac{1}{2}\cdot 2\right)=f(1),$ and we do not have a value for $f(1)$ in our table. Our input values to $g$ will need to be twice as large to get inputs for $f$ that we can evaluate. For example, we can determine $g(4)\text{.}$

We do the same for the other values to produce [link] .

[link] shows the graphs of both of these sets of points.

The graph of $g(x)$ looks like the graph of $f(x)$ horizontally compressed. Because $f(x)$ ends at $(6,4)$ and $g(x)$ ends at $(2,4),$ we can see that the $x\text{-}$ values have been compressed by $\frac{1}{3},$ because $6\left(\frac{1}{3}\right)=2.$ We might also notice that $g(2)=f\left(6\right)$ and $g(1)=f\left(3\right).$ Either way, we can describe this relationship as $g(x)=f\left(3x\right).$ This is a horizontal compression by $\frac{1}{3}.$