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Sample Test: Function I

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides a sample test on functions.

Exercise 1

Chris is 1½ years younger than his brother David. Let DD represent David’s age, and CC represent Chris’s age.

  • a. If Chris is fifteen years old, how old is David?______
  • b. Write a function to show how to find David’s age, given Chris’s age. D ( C ) = D ( C ) = size 12{D \( C \) ={}} {} ______

Exercise 2

Sally slips into a broom closet, waves her magic wand, and emerges as…the candy bar fairy! Flying through the window of the classroom, she gives every student two candy bars. Then five candy bars float through the air and land on the teacher’s desk. And, as quickly as she appeared, Sally is gone to do more good in the world.

Let ss represent the number of students in the class, and cc represent the total number of candy bars distributed. Two for each student, and five for the teacher.

  • a. Write a function to show how many candy bars Sally gave out, as a function of the number of students. c ( s ) = c ( s ) = size 12{c \( s \) ={}} {} ______
  • b. Use that function to answer the question: if there were 20 students in the classroom, how many candy bars were distributed? First represent the question in functional notation—then answer it. ______
  • c. Now use the same function to answer the question: if Sally distributed 35 candy bars, how many students were in the class? First represent the question in functional notation—then answer it. ______

Exercise 3

The function f ( x ) = f ( x ) = size 12{f \( x \) ={}} {} is “Subtract three, then take the square root.”

  • a. Express this function algebraically, instead of in words: f ( x ) = f ( x ) = size 12{f \( x \) ={}} {} ______
  • b. Give any three points that could be generated by this function:______
  • c. What x x-values are in the domain of this function?______

Exercise 4

The function y(x)y(x) is “Given any number, return 6.”

  • a. Express this function algebraically, instead of in words: y ( x ) = y ( x ) = size 12{y \( x \) ={}} {} ______
  • b. Give any three points that could be generated by this function:______
  • c. What xx-values are in the domain of this function?______

Exercise 5

z ( x ) = x 2 6x + 9 z ( x ) = x 2 6x + 9 size 12{z \( x \) =x rSup { size 8{2} } - 6x+9} {}

  • a. z(1)=z(1)= size 12{z \( - 1 \) ={}} {}______
  • b. z(0)=z(0)= size 12{z \( 0 \) ={}} {} ______
  • c. z(1)=z(1)= size 12{z \( 1 \) ={}} {}______
  • d. z(3)=z(3)= size 12{z \( 3 \) ={}} {}______
  • e. z(x+2)=z(x+2)= size 12{z \( x+2 \) ={}} {}______
  • f. z(z(x))=z(z(x))= size 12{z \( z \( x \) \) ={}} {}______

Exercise 6

Of the following sets of numbers, indicate which ones could possibly have been generated by a function. All I need is a “Yes” or “No”—you don’t have to tell me the function! (But go ahead and do, if you want to…)

  • a. (2,4)(1,1)(0,0)(1,1)(2,4)(2,4)(1,1)(0,0)(1,1)(2,4) size 12{ \( - 2,4 \) \( - 1,1 \) \( 0,0 \) \( 1,1 \) \( 2,4 \) } {}
  • b. (4,2)(1,1)(0,0)(1,1)(4,2)(4,2)(1,1)(0,0)(1,1)(4,2) size 12{ \( 4, - 2 \) \( 1, - 1 \) \( 0,0 \) \( 1,1 \) \( 4,2 \) } {}
  • c. (2,π)(3,π)(4,π)(5,1)(2,π)(3,π)(4,π)(5,1) size 12{ \( 2,π \) \( 3,π \) \( 4,π \) \( 5,1 \) } {}
  • d. (π,2)(π,3)(π,4)(1,5)(π,2)(π,3)(π,4)(1,5) size 12{ \( π,2 \) \( π,3 \) \( π,4 \) \( 1,5 \) } {}

Exercise 7

Make up a function involving music.

  • a. Write the scenario. Your description should clearly tell me—in words—how one value depends on another.
  • b. Name, and clearly describe, two variables. Indicate which is dependent and which is independent.
  • c. Write a function showing how the dependent variable depends on the independent variable. If you were explicit enough in parts (a) and (b), I should be able to predict your answer to part (c) before I read it.
  • d. Choose a sample number to show how your function works. Explain what the result means.

Exercise 8

Here is an algebraic generalization: for any number x x size 12{x} {} , x225=(x+5)(x5)x225=(x+5)(x5) size 12{x rSup { size 8{2} } - "25"= \( x+5 \) \( x - 5 \) } {}.

  • a. Plug x=3x=3 size 12{x=3} {} into that generalization, and see if it works.
  • b. 20×2020×20 size 12{"20" times "20"} {} is 400. Given that, and the generalization, can you find 15×2515×25 size 12{"15" times "25"} {} without a calculator? (Don’t just give me the answer, show how you got it!)

Exercise 9

Amy has started a company selling candy bars. Each day, she buys candy bars from the corner store and sells them to students during lunch. The following graph shows her profit each day in March.

Figure 1
A graph showing the profits for March.
  • a. On what days did she break even?
  • b. On what days did she lose money?

Exercise 10

The picture below shows the graph of y=xy=x size 12{y= sqrt {x} } {}. The graph starts at (0,0)(0,0) size 12{ \( 0,0 \) } {} and moves up and to the right forever.

Figure 2
A graph of the square root of x starting at (0,0).
  • a. What is the domain of this graph?
  • b. Write a function that looks exactly the same, except that it starts at the point (3,1)(3,1) size 12{ \( - 3,1 \) } {} and moves up-and-right from there.

Exercise 11

The following graph represents the graph y=f(x)y=f(x) size 12{y=f \( x \) } {}.

Figure 3
A graph of function f(x).
  • a. Is it a function? Why or why not?
  • b. What are the zeros?
  • c. For what xvalues x values is it positive?
  • d. For what xvalues x values is it negative?
  • e. Below is the same function f(x)f(x) size 12{f \( x \) } {}. On that same graph, draw the graph of y=f(x)2y=f(x)2 size 12{y=f \( x \) - 2} {}.
Figure 4
A graph of test function f(x).
  • f. Below is the same function f(x)f(x) size 12{f \( x \) } {}. On that same graph, draw the graph of y=f(x)y=f(x) size 12{y= - f \( x \) } {}.
Figure 5
A graph of simple test function f(x).

Extra credit:

Here is a cool trick for squaring a difficult number, if the number immediately below it is easy to square.

Suppose I want to find 312312 size 12{"31" rSup { size 8{2} } } {}. That’s hard. But it’s easy to find 302302 size 12{"30" rSup { size 8{2} } } {}, that’s 900. Now, here comes the trick: add 30, and then add 31. 900+30+31=961900+30+31=961 size 12{"900"+"30"+"31"="961"} {}. That’s the answer! 312=961312=961 size 12{"31" rSup { size 8{2} } ="961"} {}.

  • a. Use this trick to find 412412 size 12{"41" rSup { size 8{2} } } {}. (Don’t just show me the answer, show me the work!)
  • b. Write the algebraic generalization that represents this trick.

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