# Lesson 4

Using Function Notation to Describe Rules (Part 1)

• Let’s look at some rules that describe functions and write some, too.

### 4.1: Notice and Wonder: Two Functions

What do you notice? What do you wonder?

$$x$$ $$f(x)=10-2x$$
1 8
1.5 7
5 0
-2 14
$$x$$ $$g(x)=x^3$$
-2 -8
0 0
1 1
3 27

### 4.2: Four Functions

Here are descriptions and equations that represent four functions.

$$f(x)=3x-7\\$$

$$g(x)=3(x-7)\\$$

$$h(x)=\frac{x}{3}-7\\$$

$$k(x)=\dfrac{x-7}{3}\\$$

A. To get the output, subtract 7 from the input, then divide the result by 3.

B. To get the output, subtract 7 from the input, then multiply the result by 3.

C. To get the output, multiply the input by 3, then subtract 7 from the result.

D. To get the output, divide the input by 3, and then subtract 7 from the result.

1. Match each equation with a verbal description that represents the same function. Record your results.
2. For one of the functions, when the input is 6, the output is -3. Which is that function: $$f, g$$, $$h$$, or $$k$$? Explain how you know.
3. Which function value—$$f(x), g(x), h(x)$$, or $$k(x)$$—is the greatest when the input is 0? What about when the input is 10?

Mai says $$f(x)$$ is always greater than $$g(x)$$ for the same value of $$x$$. Is this true? Explain how you know.

### 4.3: Rules for Area and Perimeter

1. A square that has a side length of 9 cm has an area of 81 cm2. The relationship between the side length and the area of the square is a function.

1. Complete the table with the area for each given side length.

Then, write a rule for a function, $$A$$, that gives the area of the square in cm2 when the side length is $$s$$ cm. Use function notation.

side length (cm) area (cm2)
1
2
4
6
$$s$$
2. What does $$A(2)$$ represent in this situation? What is its value?
3. On the coordinate plane, sketch a graph of this function.

2. A roll of paper that is 3 feet wide can be cut to any length.

1. If we cut a length of 2.5 feet, what is the perimeter of the paper?

2. Complete the table with the perimeter for each given side length.

Then, write a rule for a function, $$P$$, that gives the perimeter of the paper in feet when the side length in feet is $$\ell$$. Use function notation.

side length (feet) perimeter (feet)
1
2
6.3
11
$$\ell$$
3. What does $$P(11)$$ represent in this situation? What is its value?
4. On the coordinate plane, sketch a graph of this function.

### Summary

Some functions are defined by rules that specify how to compute the output from the input. These rules can be verbal descriptions or expressions and equations. For example:

Rules in words:

• To get the output of function $$f$$, add 2 to the input, then multiply the result by 5.
• To get the output of function $$m$$, multiply the input by $$\frac12$$ and subtract the result from 3.

Rules in function notation:

• $$f(x) = (x + 2) \boldcdot 5$$ or $$5(x+2)$$
• $$m(x) = 3 - \frac12x$$

Some functions that relate two quantities in a situation can also be defined by rules and can therefore be expressed algebraically, using function notation.

Suppose function $$c$$ gives the cost of buying $$n$$ pounds of apples at \\$1.49 per pound. We can write the rule $$c(n) = 1.49n$$ to define function $$c$$.

To see how the cost changes when $$n$$ changes, we can create a table of values.

pounds of apples, $$n$$ cost in dollars, $$c(n)$$
0 0
1 1.49
2 2.98
3 4.47
$$n$$ $$1.49n$$

Plotting the pairs of values in the table gives us a graphical representation of $$c$$.

### Glossary Entries

• dependent variable

A variable representing the output of a function.

The equation $$y = 6-x$$ defines $$y$$ as a function of $$x$$. The variable $$x$$ is the independent variable, because you can choose any value for it. The variable $$y$$ is called the dependent variable, because it depends on $$x$$. Once you have chosen a value for $$x$$, the value of $$y$$ is determined.

• function

A function takes inputs from one set and assigns them to outputs from another set, assigning exactly one output to each input.

• function notation

Function notation is a way of writing the outputs of a function that you have given a name to. If the function is named $$f$$ and $$x$$ is an input, then $$f(x)$$ denotes the corresponding output.

• independent variable

A variable representing the input of a function.

The equation $$y = 6-x$$ defines $$y$$ as a function of $$x$$. The variable $$x$$ is the independent variable, because you can choose any value for it. The variable $$y$$ is called the dependent variable, because it depends on $$x$$. Once you have chosen a value for $$x$$, the value of $$y$$ is determined.