Inverse Functions – Final Project Option Name ______
Algebra II with Trigonometry Date ______Pd ______
In this project, you will investigate inverse functions, which combine both topics of algebra and analysis. Analysis is all about functions and their behavior, and there is a special analytical/graphical connection between a function and its inverse. Algebra uses inverse functions or inverse operations to solve equations.
I. Inverse Operation Quiz (5pts)
Throughout this course in Algebra II with Trigonometry, we have used inverse operations to undo (or ‘do the opposite of’) another operation and thereby solve an equation. After taking this quiz, check your answers with Mr. Yates.
What is the inverse of each of the following operations?
1. addition ______
2. multiplication ______
3. squaring ______
4. exponentiation (where x is the exponent) ______
5. sine ______
II. Linear Functions (20pts)
1. The simplest linear equation is y = x. Graph it. Create a square graph / window: (-10 to 10) by (-10 to 10); if you’re not sure how to do this, ask Mr. Yates. Since there is nothing being added, subtracted, multiplied, divided, squared, exponented, or trigonometried, there are no inverse operations to apply: it is its own inverse!
2. Let’s try y = 2x or—in function notation—f(x) = 2x. Graph it in the same plane.
3. What is happening to the x? The inverse function will have the OPPOSITE done to x. To find your inverse function, switch y and x, then solve for y. State your inverse function.
4. Graph this inverse function, and label both it and the original. Keep an eye out on the graph for how it is related to the original.
Let’s take a look at how inverse functions are related using formulas. First, it is important to understand that we use f-1(x) to denote the inverse of f(x). Be sure you do not think of –1 as an exponent. In this context, f-1(x) to just means the inverse of f(x). Take for example the functions and . These functions are inverses, because they “undo” one another. In other words, f(x) adds 5 to x, and f-1(x) subtracts 5 from x.
5. Consider the function f(x) = 4x – 9. Describe in words what f(x) does to the value of x.
6. In words, what would be the inverse of this process? (Hint: Be sure you consider the order that the inverse operations occur. Think about when you put on your socks and then put on your shoes. Do you take your socks off first?)
7. Now, write an equation that describes f-1(x). Use your answers to the last two parts, or just switch x and y, then solve for y.
8. Graph and label f(x).
9. Graph and label f-1(x).
10. What graphical relationship do you notice between a function and its inverse?
III. Quadratic Functions (20pts)
1. The simplest quadratic equation is y = x2. Graph it, along with the line y=x.
2. Find the inverse function. Switch y and x, then solve for y. Graph it. Does anything surprise you? Debrief with Mr. Yates here.
3. Another quadratic function is f(x) = 2(x – 3)2 – 5. What form of quadratic equation is this? Graph it, along with the line y=x.
4. Find the inverse function (always show work!). Graph it.
5. Compare the domains and ranges of f and f-1. Make a conjecture.
IV. Trigonometric Functions (20pts)
1. The simplest sine function is y = sin(x). Graph it, along with the line y=x.
2. Find the inverse function. Graph it.
3. What is the domain of y=sin(x)? What is its range? What is the domain of the inverse you found? What is the range?
4. Another sine function is f(x) = 2sin(x – 3) – 5. What are its period and amplitude? Graph it, along with the line y=x.
5. Find the inverse function (always show work!). Graph it. Compare the domains and ranges of f and f-1.
V. Exponential Functions (20pts)
1. The simplest exponential function is y = 2x. Graph it, along with the line y=x.
2. Find the inverse function. Graph it. Compare domains and ranges.
3. Another exponential function is f(x) = 3· 2x – 5. Is this exponential growth or decay? Graph it.
4. Find the inverse function (always show work!). Graph it. Compare the domains and ranges of f and f-1.
5. Has your conjecture from III-5 held up? If so, restate it on the title page! If not, revise it here. Same instructions for the relationship you noticed in II-10.
VI. The hunting of the snark – Lewis Carroll (10pts)
Lewis Carroll (Charles Lutwidge Dodgson) was an English author born in 1832. You are probably familiar with Carroll’s Alice’s Adventures in Wonderland. Not only was Carroll an accomplished author, but he also had a great interest in the field of mathematics.
The Hunting of the Snark is a nonsense poem about a strange group of people that set sail to hunt for an elusive bird, called a snark. During the story, the Beaver and the Butcher are at the same location in pursuit of the snark. During their hunt, counting to three perplexes the mentally limited Beaver. Below is an excerpt from the fifth fit (or chapter) of The Hunting of the Snark titled “The Beaver’s Lesson.”
1 The Beaver had counted with scrupulous care,
Attending to every word:
But it fairly lost heart, and outgrabe in despair,
When the third repetition occurred.
2 It felt that, in spite of all possible pains,
It had somehow contrived to lose count,
And the only thing now was to rack its poor brains
By reckoning up the amount.
3 "Two added to one--if that could but be done,"
It said, "with one's fingers and thumbs!"
Recollecting with tears how, in earlier years,
It had taken no pains with its sums.
4 "The thing can be done," said the Butcher, "I think.
The thing must be done, I am sure.
The thing shall be done! Bring me paper and ink,
The best there is time to procure."
5 The Beaver brought paper, portfolio, pens,
And ink in unfailing supplies:
While strange creepy creatures came out of their dens,
And watched them with wondering eyes.
6 So engrossed was the Butcher, he heeded them not,
As he wrote with a pen in each hand,
And explained all the while in a popular style
Which the Beaver could well understand.
7 "Taking Three as the subject to reason about--
A convenient number to state--
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.
8 "The result we proceed to divide, as you see,
By Nine Hundred and Ninety Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.
1. The Butcher started with the number three, and then performed a number of operations in stanzas 7 and 8 to result in the number three. Consider the seventh stanza above. What happens to the number three in this stanza?
2. Suppose rather than the number 3, we use the variable x. Write a function f(x) that describes what operations occur in stanza 7.
3. Now consider the eighth stanza. What happens to our result from before?
4. Suppose we use the variable x once again to write a function g(x) that describes the operations that occur in stanza 8.
5. What is the relationship between f(x) and g(x)? Prove it!
VII. Cover page (5pts)
Should include name, date, title, and the two parts mentioned in V-5.