From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# High School: Functions

### Interpreting Functions HSF-IF.A.3

3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

When handling functions (careful, they're fragile!), it's easy to see patterns emerge when comparing the x and y values to each other. Students should know that these patterns are not coincidences and, unlike certain patterned wallpaper, they won't make your kitchen look like it belongs in the 1970s.

Students should know that these patterns can be thought of as sequences, or a list of numbers. Sequences can be either arithmetic (where the same number is added or subtracted) or geometric (where the same number is multiplied or divided).

Arithmetic sequences can be converted into functions of the form A(n) = A(1) + (n – 1)d where A(n) is the value of the nth term, A(1) is the value of the first term, n is the term number, and d is the common difference between the terms. So the sequence 3, 8, 13, 18... can be thought of as the function A(n) = 3 + 5(n – 1). That way, the nth term of the sequence will have the value A(n).

Geometric series take a similar form: G(n) = G(1) × rn – 1 where G(n) is the value of the nth term, G(1) is the value of the first term, n is the term number, and r is the common ratio between the terms. So the sequence 3, 6, 12, 24, 48... can be thought of as the function G(n) = 3 × 2n – 1. That way, the nth term of the sequence will have the value G(n).

Students should know that sequences can be defined recursively, or using previous terms to define future terms. For instance, The Fibonacci sequence is a list of numbers where each term is the sum of the two before it. As such, we end up with 1, 1, 2, 3, 5, 8, 13, and so on. We can define this recursively as f(n + 1) = f(n) + f(n – 1). Once the first two terms are defined, (f(0) = f(1) = 1), the sequence can just keep on going, not unlike the Energizer bunny.

Students should be so skilled in seeing the functions in sequences that you should find them accidentally converting other things into functions, too. In fact, it should become second nature.

We're talking analyzing the number of breaths they take as a function of time. The number of times they sneeze as a function of their allergies. The number of detentions they get as a function of how often they kick soccer balls at Coach Gibson. Sorry, Coach.

#### Drills

1. What is the tenth number of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...)?

55

The tenth term of the Fibonacci sequence is the sum of the values of the eighth and ninth terms. If we continue writing out the sequence, we'll get 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. All we really need to do is count the number of terms, and we'll have 55 as the tenth term.

2. What are the next two terms in the sequence 1, 4, 7...?

10 and 13

The pattern is to add 3 to the preceding number. Hopefully that's pretty clear. The next two terms are 7 + 3 = 10 and 10 + 3 = 13. So our answer is (B).

3. What are the next two terms in the sequence 12, 48, 192...?

768 and 3,072

The pattern is to multiply the preceding number by 4. We know that because 12 × 4 = 48 and 48 × 4 = 192. So our next numbers will be 192 × 4 = 768 and 768 × 4 = 3,072. Also known as (D).

4. What is the correct function for the sequence -12, -11.75, -11.5, -11.25...?

f(n) = -12.25 + 0.25n

This is clearly an arithmetic sequence, and all our answer choices are more or less in that form. Substituting in our first term (-12) and our common difference (0.25), we should get f(n) = -12 + 0.25(n – 1). If we distribute the 0.25 and simplify, we'll get (C). Answer (A) works only if our first term is -11.75 (since f(1) = -12 + 0.25 = -11.75), and (B) subtracts 0.25 rather than adding it. Answer (D) is wrong entirely because it switches the first term for the common difference.

5. A sequence has f(1) = 12 and f(3) = 34. If f(2) is the average of f(1) and f(3), which function correctly describes the sequence?

f(n) = 1 + 11n

Our first order of business is to find f(2). The average of 12 and 34 is 23. Our sequence is now 12, 23, 34. To move from one term to the next, we add 11. While (B) seems like the right choice at first glance, actually plugging in n = 1 gives us f(1) = 34 because in parenthesis, we have n + 1 rather than n – 1. The right choice is (A).

6. What is the correct function for the sequence 1, -2, 3, -4, 5, -6...?

f(n) = n(-1)n – 1

We know that f(n) has the same value as n. The only part that changes is the negative sign. This can't be arithmetic because we're not adding or subtracting anything consistently, but we are multiplying each term by -1. Our function has to be either (B) or (D), but solving for f(1) gives -1 for (B) and 1 for (D). Since we start with 1, (D) is the right answer.

7. What is the correct function for the sequence 256, 128, 64, 32, 16...?

f(n) = 256 × 21 – n

You might think this is unnecessarily complicated, and you're right. All we're doing is dividing by 2, so why do these functions look so complicated? Bear with us. Dividing by 2 is the same as multiplying by 0.5. That might point your eye to (D), but be careful. A good way to test these functions is simply to plug in some n value and check it against the original sequence (and for (D), f(2) = 64, which isn't the case for our function). It should be obvious that (A) and (B) won't work, but why is (C) right? The n is negative in the exponent, which translates to the reciprocal of the base. So when n = 2, we'll end up with . This is exactly what we want.

8. Which of the following sequences is expressed by the function f(n) = 1 × 5n?

5, 25, 125, 625...

Somehow, we're guessing that f(1) = 5, maybe because every answer choice's first term is 5. So finding f(1) won't tell us much, but f(2) sure will. We can calculate the second term to be f(2) = 1 × 52 = 1 × 25 = 25. We could also realize that the function in the problem is of the geometric sequence variety, eliminating (B) immediately. Multiplying by a ratio of 5, gets rid of (B) and (D) as well, leaving us with the right answer: (A).

9. What is the seventeenth term of the series 1, 3, 7, 15, 31...?

131,071

While it might be difficult to come up with a function for this sequence, it's much better than mindlessly calculating twelve more numbers. We think so, anyway. At first, you might think it's f(n) = 1 + 2n – 1, but that fails once we get to f(3) = 5. Instead, compare the sequence to simply f(n) = 2n, or 2, 4, 8, 16, 32... Our sequence is just that exact sequence minus 1, or f(n) = 2n – 1. Now, calculating the seventeenth term shouldn't be a problem: f(17) = 217 – 1 = 131,072 – 1 = 131,071. Done and done.

10. Which of the following is a possible definition of the sequence 4, 6, 9, 13.5, 20.25... given that f(1) = 4?