audio read-through More Practice with Function Notation

Want some basic practice with functions first?

Recall from Introduction to Function Notation that a function is a rule that takes an input, does something to it, and gives a unique corresponding output.

There is a special notation (called ‘function notation’) that is used to represent this situation: if the function name is $\,f\,,$ and the input name is $\,x\,,$ then the unique corresponding output is called $\,f(x)\,.$ The notation ‘$f(x)\,$’ is read aloud as:  ‘$\,f\,$  of  $\,x\,$’.

So, what exactly is $\,f(x)\,$?   Answer: It is the output from the function $\,f\,$ when the input is $\,x\,\,.$

This exercise gives more advanced practice with function notation.


Question: Let $\,f(x) = x^2 + 2x\,.$ Find and simplify:  $f(-3)$
Solution: $$ \begin{align} \cssId{s20}{f(-3)}\ &\cssId{s21}{= (-3)^2 + 2(-3)}\cr &\cssId{s22}{= 9 - 6}\cr &\cssId{s23}{= 3} \end{align} $$
Question: Let $\,f(x) = x^2 + 2x\,.$ Find and simplify:  $f(x+1)$
Solution: $$ \begin{align} \cssId{s28}{f(x+1)}\ &\cssId{s29}{= (x+1)^2 + 2(x+1)}\cr &\cssId{s30}{= x^2 + 2x + 1 + 2x + 2}\cr &\cssId{s31}{= x^2 + 4x + 3} \end{align} $$

The ‘Empty Parentheses Method’

Some people find it helpful to use the so-called ‘empty parentheses method’ to help with function evaluation.

For example, take the function rule $\,f(x) = x^2 + 2x\,$ and rewrite it as: $$\cssId{s35}{f(\text{blah})} \cssId{s36}{= (\text{blah})^2 + 2(\text{blah})}$$

Or, even more simply, just leave a blank space for the input—a pair of empty parentheses where the input should be: $$\cssId{s39}{f(\ \ \ \ )} \cssId{s40}{= (\ \ \ \ )^2 + 2(\ \ \ \ )}$$

Then, when you want to find (say) $\,f(x+1)\,,$ just put the input, $\,x+1\,,$ inside every set of empty parentheses: $$\cssId{s43}{f(x+1)} \cssId{s44}{= (x+1)^2 + 2(x+1)}$$ Voila!

Question: Let $\,f(x) = 5\,.$ Find and simplify:  $f(x+1)$
Solution: The function $\,f\,$ is a constant function: no matter what the input is, the output is $\,5\,.$ That is, $\,f(\text{anything}) = 5\,.$ So, $\,f(x+1) = 5\,.$
Question: Let $\,f(x) = x^2 - 2x\,.$ Find and simplify:  $f(1) + f(3)$
Solution: $$ \begin{align} &\cssId{s58}{f(1) + f(3)}\cr &\qquad\cssId{s59}{= \overset{f(1)}{\overbrace{(1^2 - 2\cdot 1)}}} \cssId{s60}{+ \overset{f(3)}{\overbrace{(3^2 - 2\cdot 3)}}}\cr\cr &\qquad\cssId{s61}{= (1 - 2) + (9- 6)}\cr\cr &\qquad\cssId{s62}{= 2} \end{align} $$

Concept Practice