Task Sets: Recognize sequences as functions - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

A function assigns each element of the domain exactly one element of the range. Which of the following is a function?

A.

A rule that maps 3 to both 7 and 9.

B.

The relationship $\{(1, 4), (2, 7), (3, 10)\}$ .

C.

A rule where input 5 has no output.

D.

A rule that maps every input to 0 outputs.

2.

The sequence 5, 8, 11, 14, 17, ... is graphed on the coordinate plane with $n$ on the horizontal axis. Which graph is correct?

A.

A solid straight line through the plotted points.

B.

Isolated dots at $(1,5)$ , $(2,8)$ , $(3,11)$ , $(4,14)$ , $(5,17)$ with no line connecting them.

C.

A curve connecting all plotted points.

D.

Points at $(5,1)$ , $(8,2)$ , $(11,3)$ , $(14,4)$ , $(17,5)$ .

3.

The function $f(x) = 2x + 1$ has domain $\{1, 2, 3, 4\}$ . What is $f(3)$ ?

A.

5

B.

6

C.

7

D.

8

B

Fluency Practice

1.

The sequence $a(n) = 3n - 1$ has domain $\{1, 2, 3, 4, 5, ...\}$ . Find $a(6)$ .

2.

The sequence 4, 7, 10, 13, 16, ... is what type?

A.

Geometric, because consecutive terms have a constant ratio.

B.

Arithmetic, because consecutive terms have a constant difference of 3.

C.

Neither arithmetic nor geometric.

D.

Geometric, because the terms are always increasing.

3.

The arithmetic sequence has first term $a(1) = 5$ and common difference $d = 4$ . Use the explicit formula $a(n) = a_1 + d(n-1)$ to find $a(10)$ .

4.

The geometric sequence has first term $a(1) = 2$ and common ratio $r = 3$ . Use the explicit formula $a(n) = a_1 \cdot r^{n-1}$ to find $a(4)$ .

5.

Which of the following is a complete recursive definition for the sequence 2, 5, 8, 11, ...?

A.

$a(n) = a(n-1) + 3$ for $n \geq 2$ .

B.

$a(1) = 2; \; a(n) = a(n-1) + 3$ for $n \geq 2$ .

C.

$a(n) = 3n - 1$ .

D.

$a(1) = 2; \; a(n) = 3a(n-1)$ for $n \geq 2$ .

C

Varied Practice

D

Word Problems

1.

A theater has 20 seats in row 1 and adds 3 seats per row. The number of seats in row $n$ is $s(n) = 20 + 3(n - 1)$ .

1.

How many seats are in row 8?

2.

Write a recursive definition for $s(n)$ . Which option is complete and correct?

A.

$s(n) = s(n-1) + 3$ for $n \geq 2$ .

B.

$s(1) = 20; \; s(n) = s(n-1) + 3$ for $n \geq 2$ .

C.

$s(1) = 20; \; s(n) = 3s(n-1)$ for $n \geq 2$ .

D.

$s(n) = 20 + 3(n-1)$ .

2.

A ball is dropped from a height of 80 cm. Each bounce reaches 75% of the previous height. The height after $n$ bounces is $h(n) = 80 \cdot (0.75)^n$ .

What is the height (in cm) after 2 bounces? Round to the nearest whole number.

E

Error Analysis

1.

Student work: "The sequence 2, 5, 8, 11, 14, ... is just a pattern — not a function. Functions need equations like $f(x) = 2x + 1$ , not just a list of numbers."

Is the student correct? Identify the error.

A.

The student is correct — sequences are patterns, not functions.

B.

The student is wrong. The sequence is a function: input $n = 1$ gives output 2, input $n = 2$ gives output 5, etc. Each integer input has exactly one output, satisfying the function definition.

C.

The student is partially correct — the sequence needs an explicit formula before it can be called a function.

D.

The student is wrong only because sequences must use the notation $a(n)$ to be functions.

2.

Student's recursive definition for the sequence 3, 9, 27, 81, ...:

" $a(n) = 3 \cdot a(n-1)$ for $n \geq 2$ ."

What is missing, and how does it affect the definition?

F

Challenge / Extension

1.

The Fibonacci sequence is defined by $f(0) = 1$ , $f(1) = 1$ , $f(n+1) = f(n) + f(n-1)$ .

If you change the initial conditions to $f(0) = 2$ and $f(1) = 3$ , what is the value of $f(5)$ for this new sequence?

2.