1.19 Sample test: functions ii

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 $L$ be the radius of Lisa’s cookies. Let $J$ be the radius of Joe’s cookies. Let $a$ be the area of Joe’s cookies.

• A

  Write a function $J(L)$ that shows the radius of Joe’s cookies as a function of the radius of Lisa’s cookies.

• B

  Write a function $a(J)$ 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 $a(J(L))$ 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 $49\pi$ square inches in area, what was the radius of Lisa’s cookies? First, write the question in function notation—then solve it.

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.

Here is the algorithm for converting the temperature from Celsius to Fahrenheit. First, multiply the Celsius temperature by $\frac{9}{5}$ . Then, add 32.

• A

  Write this algorithm as a mathematical function: Celsius temperature (C) goes in , Fahrenheit temperature $(F)$ comes out . $F=(C)$ ______

• 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).

$f(x)=\sqrt{x+1}$ . $g(x)=\frac{1}{x}$ . For (a)-(e), I am not looking for answers like ${\left[g(x)\right]}^2$ . Your answers should not have a $g$ or an $f$ in them, just a bunch of " $x$ "’s.

• A

  $f(g(x))=$

• B

  $g(f(x))=$

• C

  $f(f(x))=$

• D

  $g(g(x))=$

• E

  $g(f(g(x)))=$

• F

  What is the domain of $f(x)$ ?

• G

  What is the domain of $g(x)$ ?

$f(x)=\text{20}-x$

• 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 .)

$f(x)=3+\frac{x}{7}$

• A

  What is the domain?

• B

  What is the inverse function?

• C

  Test your inverse function. (Same note as above.)

$f(x)=\frac{\mathrm{2x}}{\mathrm{3x}-4}$

• A

  What is the domain?

• B

  What is the inverse function?

• C

  Test your inverse function. (Same note as above.)

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

$\mathrm{6x}+\mathrm{3y}=\text{10}$

$y=\text{mx}+b$ format:___________

Slope:___________

$y$ -intercept:___________

Graph it!