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Sample Test: Functions II

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

Summary: This module provides a second sample test on functions.

Exercise 1

Joe and Lisa are baking cookies. Every cookie is a perfect circle. Lisa is experimenting with cookies of different radii (*the plural of “radius”). Unknown to Lisa, Joe is very competitive about his baking. He sneaks in to measure the radius of Lisa’s cookies, and then makes his own cookies have a 2" bigger radius.

Let LL size 12{L} {} be the radius of Lisa’s cookies. Let JJ size 12{J} {} be the radius of Joe’s cookies. Let aa size 12{a} {} be the area of Joe’s cookies.

  • a. Write a function J(L)J(L) size 12{J \( L \) } {} that shows the radius of Joe’s cookies as a function of the radius of Lisa’s cookies.
  • b. Write a function a(J)a(J) size 12{a \( J \) } {} that shows the area of Joe’s cookies as a function of their radius. (If you don’t know the area of a circle, ask me—this information will cost you 1 point.)
  • c. Now, put them together into the function a(J(L))a(J(L)) size 12{a \( J \( L \) \) } {} that gives the area of Joe’s cookies, as a direct function of the radius of Lisa’s.
  • d. Using that function, answer the question: if Lisa settles on a 3" radius, what will be the area of Joe’s cookies? First, write the question in function notation—then solve it.
  • e. Using the same function, answer the question: if Joe’s cookies end up 49π49π square inches in area, what was the radius of Lisa’s cookies? First, write the question in function notation—then solve it.

Exercise 2

Make up a word problem involving composite functions, and having something to do with drug use. (*I will assume, without being told so, that your scenario is entirely fictional!)

  • a. Describe the scenario. Remember that it must have something that depends on something else that depends on still another thing. If you have described the scenario carefully, I should be able to guess what your variables will be and all the functions that relate them.
  • b. Carefully name and describe all three variables.
  • c. Write two functions. One relates the first variable to the second, and the other relates the second variable to the third.
  • d. Put them together into a composite function that shows me how to get directly from the third variable to the first variable.
  • e. Using a sample number, write a (word problem!) question and use your composite function to find the answer.

Exercise 3

Here is the algorithm for converting the temperature from Celsius to Fahrenheit. First, multiply the Celsius temperature by 95 9 5 . Then, add 32.

  • a. Write this algorithm as a mathematical function: Celsius temperature (C) goes in, Fahrenheit temperature (F)(F) size 12{ \( F \) } {} comes out. F=(C)F=(C) size 12{F= \( C \) } {}______
  • b. Write the inverse of that function.
  • c. Write a real-world word problem that you can solve by using that inverse function. (This does not have to be elaborate, but it has to show that you know what the inverse function does.)
  • d. Use the inverse function that you found in part (b) to answer the question you asked in part (c).

Exercise 4

f ( x ) = x + 1 f ( x ) = x + 1 size 12{f \( x \) = sqrt {x+1} } {} . g(x)=1xg(x)=1x size 12{g \( x \) = { {1} over {x} } } {}. For (a)-(e), I am not looking for answers like g(x)2g(x)2 size 12{ left [g \( x \) right ] rSup { size 8{2} } } {}. Your answers should not have a gg size 12{g} {} or an ff size 12{f} {} in them, just a bunch of " xx size 12{x} {}"’s.

  • a. f ( g ( x ) ) = f ( g ( x ) ) = size 12{f \( g \( x \) \) ={}} {}
  • b. g ( f ( x ) ) = g ( f ( x ) ) = size 12{g \( f \( x \) \) ={}} {}
  • c. f ( f ( x ) ) = f ( f ( x ) ) = size 12{f \( f \( x \) \) ={}} {}
  • d. g ( g ( x ) ) = g ( g ( x ) ) = size 12{g \( g \( x \) \) ={}} {}
  • e. g ( f ( g ( x ) ) ) = g ( f ( g ( x ) ) ) = size 12{g \( f \( g \( x \) \) \) ={}} {}
  • f. What is the domain of f(x)f(x) size 12{f \( x \) } {}?
  • g. What is the domain of g(x)g(x) size 12{g \( x \) } {}?

Exercise 5

f ( x ) = 20 − x f ( x ) = 20 − x size 12{f \( x \) ="20" - x} {}

  • a. What is the domain?
  • b. What is the inverse function?
  • c. Test your inverse function. (No credit for just the words “it works”—I have to see your test.)

Exercise 6

f ( x ) = 3 + x 7 f ( x ) = 3 + x 7 size 12{f \( x \) =3+ { {x} over {7} } } {}

  • a. What is the domain?
  • b. What is the inverse function?
  • c. Test your inverse function. (Same note as above.)

Exercise 7

f ( x ) = 2x 3x − 4 f ( x ) = 2x 3x − 4 size 12{f \( x \) = { {2x} over {3x - 4} } } {}

  • a. What is the domain?
  • b. What is the inverse function?
  • c. Test your inverse function. (Same note as above.)

Exercise 8

For each of the following diagrams, indicate roughly what the slope is.

Figure 1: a.
A graph with a steep negative slope.
Figure 2: b.
A graph with a slight positive slope.
Figure 3: c.
A graph with a zero slope.

Exercise 9

6x + 3y = 10 6x + 3y = 10 size 12{6x+3y="10"} {}

y=mx+by=mx+b size 12{y= ital "mx"+b} {} format:___________

Slope:___________

yy size 12{y} {}-intercept:___________

Graph it!

Extra credit:

Two numbers have the peculiar property that when you add them, and when you multiply them, you get the same answer.

  • a. If one of the numbers is 5, what is the other number?
  • b. If one of the numbers is xx size 12{x} {}, what is the other number? (Your answer will be a function of xx size 12{x} {}.)
  • c. What number could xx size 12{x} {} be that would not have any possible other number to go with it?

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