Lesson 6 Different Combinations Practice Understanding

Learning Focus

Combine functions defined by graphs or tables.

How can I represent function addition and multiplication? How are these operations defined when the functions being added or multiplied are defined by graphs or tables?

Open Up the Math: Launch, Explore, Discuss

We have found the value of being able to combine different function types in various ways to model a variety of situations. In this task you will practice combining functions when they are described in different ways: graphically, numerically, or algebraically.

1.

a.

Add the following two functions together graphically. That is, do not write the algebraic rules for each individual function, add them together, and then graph the result. See if you can produce the resulting graph by just working with the points on the two graphs and considering what happens when two functions are combined using the operation of addition.

b.

Which points are most helpful in determining the shape of the resulting graph, and why?

2.

a.

Multiply the following two functions together graphically. That is, do not write the algebraic rules for each individual function, multiply them together, and then graph the result. See if you can produce the resulting graph by just working with the points on the two graphs and considering what happens when two functions are combined using the operation of multiplication.

b.

Which points are most helpful in determining the shape of the resulting graph, and why?

3.

We have used the following type of diagram to illustrate function composition. Draw a similar type of diagram to illustrate what happens when two functions are combined by addition or multiplication. Your diagram should clearly show how the output values are obtained for specific input values.

4.

Functions $f$ and $g$ are defined numerically in the following table. No other points exist for these functions other than the points given. Find the output values for each of the other combinations of functions indicated. Fill in as many points as are defined based on the given data. Use the same input values for all functions.

$x$	$f (x)$	$g (x)$
$0$	$0$	$- 3$
$1$	$2$	$- 2$
$2$	$4$	$- 1$
$3$	$6$	$0$
$4$	$8$	$1$
$5$	$10$	$2$
$6$	$12$	$3$
$7$	$14$	$4$
$8$	$16$	$5$

5.

Remember the race between the tortoise and the hare? Well, their friends and families have come to cheer them on, and have positioned themselves at various places along the course. Because hares are quick and eager to know the outcome of the race, more of them have congregated towards the end of the course. Because turtles are slow and more anxious to cheer their champion off to a good start, more of them have congregated at the beginning of the race. In fact, the density (or number of animals/meter) of turtles and rabbits along the course as a function of the distance from the starting line is given by the following functions.

The tortoise: $a (d) = 243 \cdot {(\frac{1}{3})}^{\frac{1}{20} d}$ ( $a$ is in turtles per meter, $d$ in meters)

The hare: $a (d) = 2^{\frac{1}{10} d}$ ( $a$ is in rabbits per meter, $d$ in meters)

The distance from the starting line, as a function of the elapsed time since the start of the race, is given for the tortoise and the hare by the following functions.

The tortoise: $d (t) = 2^{t}$ ( $d$ in meters, $t$ in seconds)

The hare: $d (t) = t^{2}$ $(d$ in meters, $t$ in seconds)

The tortoise and the hare are anxious to know how many of their friends and family they are passing at any instant in time along the race.

Interpreting the functions:

a.

In the tortoise equation, what do $243$ , $\frac{1}{3}$ , and $\frac{1}{20} d$ mean in terms in the context?

b.

In the hare equation, what does $\frac{1}{10} d$ mean in terms of the context?

c.

If the race is $100 meters$ long, create functions for the tortoise and for the hare that will calculate the number of turtles or rabbits they will pass at any time, $t$ , after the race begins. Include a reasonable or restricted domain for each function.

d.

If the race is $100 meters$ long, create a function that will tell how many spectators (rabbits and turtles) are watching at any distance away from the start of the race.

e.

Who is passing the most friends and family, the tortoise or the hare, $5 seconds$ after the race begins?

Ready for More?

The following graph is the sum of two trigonometric functions. See if you can find the two individual functions that were added together to produce this graph.

Takeaways

To add two functions $f (x)$ and $g (x)$ graphically:

To multiply two functions $f (x)$ and $g (x)$ graphically:

Strategic values for $x$ include:

To add or multiply two functions defined by a table:

To compose two functions defined by a table:

To add or multiply two functions defined by algebraic expressions:

To compose two functions $f (g (x))$ defined by algebraic expressions:

A restricted domain defines:

Vocabulary

restricted domain
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to graph the sum or product of two functions which are only defined by their graphs. We also learned how to complete a table of values for the sum, product, inverse or composition of functions defined only by a table. We also modeled a context by selecting whether we should use a composition function or a combination function.

Retrieval

1.

An insurance company compiled the ages of the $145$ teachers in a school. The ages ranged from $23$ to $65$ and were grouped in intervals as shown in the table.

Age Interval	Midpoint of Interval	Frequency of Interval
$20 – 30$	$25$	$18$
$30 – 40$	$35$	$37$
$40 – 50$	$45$	$45$
$50 – 60$	$55$	$32$
$60 – 70$	$65$	$13$

Make a histogram of the grouped data in the chart. (Note: The midpoint of each cell is given in the horizontal axis. The sides of the bins will match the age interval. Frequency is the vertical height.)

2.

Use the laws of logarithms to write $\log \frac{3 x (x - 2)}{{(x + 5)}^{2}}$ as the sum or difference of logarithms. Express powers as factors.

3.

Write $3 \log u + 4 \log v - \log w$ as a single logarithm.