Lesson 5 Translating My Composition Solidify Understanding

Learning Focus

Write and evaluate composite functions.

What can I learn about a composite function by attending to its component parts?

Open Up the Math: Launch, Explore, Discuss

All this work with modeling rides and waiting lines at the local amusement park may have you wondering about the variety of ways of combining functions. In this task we continue building new functions from old, familiar ones.

Suppose you have the following “starter set” of functions.

$f (x) = x + 5$

$g (x) = x^{2}$

$h (x) = 3 x$

$j (x) = 2^{x}$

$k (x) = x - 1$

$m (x) = \sin (x)$

$n (x) = \cos (x)$

You and a partner will then do the following steps with your given set of functions:

Build a composite function using any three of the above function rules in any order.
Write your final function rule as a single algebraic expression in terms of $x$ .
Give your function rule to another partnership. You should also receive a function rule from them.
You and your partner should fill in the following diagram, decomposing your rule into its component parts and combining them in the correct order.

1.

First, let’s try this example:

You and your partner are given $f_{1} (x) = 3 {(x + 5)}^{2}$ . Complete this diagram to decompose this composition into its component parts.

2.

To test your decomposition, you can select an input for the first function and follow the instructions for each function to get the final output . See if you get the same results as when you evaluate the function rule for the same numbers. This is particularly helpful if the form of the function equation was modified in any way during step 2. What do you notice when you do this?

3.

Now it’s your turn! You and your partner should create your own function rule using the set of functions given at the beginning of this task and following the four steps given above. Another partnership should do the same and give you their function rule.

Record the function rule you received here:

Complete this diagram to decompose your partner’s composition into its component parts.

Test your decomposition for a few values. Make any adjustments necessary based on your test results.

Try out a few more examples with your partner. You might try using four, or even all five functions in your composition function. You can also use repetition of functions, such as, $g (f (g (x)))$ .

4.

Instead of giving you the function rule, suppose your partner gives you the following input-output table. (Note: The function represented by this table was created from the set of starter rules given at the beginning of this lesson.)

a.

Can you create the composition function rule based on this information? Describe how you used the numbers in this table to create your rule.

$x$	$f_{1} (x)$
$- 1$	$5 \frac{1}{2}$
$0$	$6$
$1$	$7$
$2$	$9$
$3$	$13$
$4$	$21$

b.

Now find the composition function rule for this table.

$x$	$f_{2} (x)$
$0$	$5 \frac{1}{2}$
$1$	$6$
$2$	$7$
$3$	$9$
$4$	$13$
$5$	$21$

5.

Is function composition commutative? Give reasons to support your answer.

Ready for More?

Examine your answer to the question, “Is function composition commutative?” from the Ready for More in the previous lesson by finding cases when it is commutative and when it is not. Can you generalize under what conditions function composition is commutative?

Takeaways

When given a composition function rule, it can be useful to decompose it into its component parts, and represent it symbolically as the composition of more than one function. Strategies I can use to decompose a function rule are:

When given a table of values for a composite function, we might be able to identify one of the functions in the composition by looking for characteristics of the parent function in the data by checking for:

Vocabulary

decomposition of functions
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we decomposed complex function rules into sequences of smaller functions that could be evaluated to give the same results as more complicated functions.

Retrieval

1.

Use the table to find the indicated function values.

$x$	$f (x)$	$g (x)$
$- 3$	$8$	$- 2$
$- 2$	$5$	$1$
$- 1$	$1$	$2$
$0$	$2$	$15$
$1$	$- 3$	$- 1$
$2$	$0$	$3$
$3$	$15$	$0$

$f (g (2)) =$

$f (g (- 1)) =$

$g (f (2)) =$

$g (g (- 3)) =$

2.

Given the graph of $y = \log_{3} x$ , graph:

$y = 2 + \log_{3} x$
$y = 1 - \log_{3} x$
$y = \log_{3} (x + 3)$