New functions from old [1.8]

Composite functions, and function transformations

Composite functions

Let's say we have a table of values of two functions of $x$:

How do you evaluate: $$f(g(2)) ?$$

Work your way from the inside out!

1. Start with the inside function:
   $g(2)=\color{blue}{5}$
   
2. Working your way out:
   $f(\color{blue}{5}\color{black})=11$
   
   So... $f(g(2))=11$ .

   another example

   Given $f(x)=x^3$ and $u(t)=t-1$...

   $f(u(t))=f(t-1)=(t-1)^3$ and...

   $u(f(x))=u(x^3)=x^3-1$.
   
   Zhydrac

   Transformations

   • $a$ - vertical stretching / shrinking
   • $b$ - horizontal stretching / shrinking
   • $c$ - shift horizontally
   • $d$ - shift vertically

   Think about the order: $$6+4*2 \stackrel{?}{=} 4*2+6\nonumber$$ Without parentheses, always

   1. multiply first,
   2. then add.

      Similarly, $$a+c*f(x)$$ always

      1. muliply: do the vertical stretch first,
      2. then add...the vertical shift.

      Alternatively, tweak this a bit to $a*f(\,b(x-c)\,)+d$ in order to stretch first, then shift horizontally. Sometimes that's more convenient.

   3. Practicing... (.ppt) refers to today's class handout.