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.

      Blank grid. Horizontal axis, 0 to 8, side length in centimeters. Vertical axis, 0 to 50 by 10’s, area in centimeters squared.
  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?

      Roll of paper, slightly unrolled. Its width is 3 feet.
    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.

      Blank grid. Horizontal axis, 0 to 8, side length in feet. Vertical axis, 0 to 25 by 5’s, perimeter in feet.

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 \(c\).

Points plotted. Horizontal axis, 0 to 8, n, pounds of apples. Vertical axis, 0 to 10, cost in dollars. Points at 0 comma 0, approximately 1 comma 1 point 5, 2 comma 3, 3 comma 4 point 5, 4 comma 6.

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.