Task Sets: Determine explicit or recursive expressions - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

If $f(x) = 3x - 1$ , what is $f(4)$ ?

A.

11

B.

13

C.

9

D.

12

2.

A sequence is defined as a function with what type of domain?

A.

All real numbers

B.

Positive real numbers

C.

Integer domain (counting numbers)

D.

Rational numbers only

3.

The sequence $a_n$ has $a_1 = 10$ and each term is 4 less than the previous. What is $a_3$ ?

B

Fluency Practice

1.

A parking garage charges $5 to enter plus $3 per hour. The explicit formula for the total cost is $C(h) = 3h + 5$ . Find $C(7)$ .

2.

A bank account starts with $2000 and grows by 4% per year. The explicit formula is $V(t) = 2000(1.04)^t$ . Find $V(3)$ . Round to the nearest cent.

3.

A sequence starts at $f(0) = 6$ . Each term is 5 more than the previous. Which is a correct recursive definition?

A.

$f(0) = 6$ , $f(n) = f(n-1) + 5$

B.

$f(n) = f(n-1) + 5$ (no initial value stated)

C.

$f(0) = 6$ , $f(n) = 5n$

D.

$f(0) = 6$ , $f(n) = f(n) + 5$

4.

A recursive sequence is defined by $f(0) = 3$ and $f(n) = f(n-1) \cdot 2$ . What is $f(4)$ ?

5.

The explicit formula $f(n) = 4n + 7$ corresponds to a recursive definition with $f(0) = 7$ . What value is added at each step (the common difference)?

C

Varied Practice

1.

A financial analyst needs to find the value of a savings account after exactly 40 years. Which representation is most efficient for this purpose?

A.

Recursive — it naturally models how interest compounds year by year

B.

Explicit — it lets you compute the value at year 40 directly without computing previous years

C.

Procedural — a step-by-step description is always clearest

D.

Either form — they produce the same result so efficiency doesn't matter

2.

The recursive definition $f(0) = 500$ , $f(n) = f(n-1) \cdot 1.05$ represents an exponential function. The equivalent explicit formula is $f(n) = 500 \cdot (\text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}})^n$ , and the common ratio is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

base of the explicit formula:

common ratio:

3.

A shipping company charges $15 for items up to 5 lb, then $3 for each additional pound, plus $0.10 per mile of delivery. Write numbered steps for computing the total cost. Then explain in one sentence why this procedural description defines a valid function.

4.

A recursive sequence is defined by $g(0) = 2$ and $g(n) = g(n-1) \cdot 3$ . A student wants $g(50)$ and writes: " $g(50) = g(49) + 3 = 50 + 3 = 53$ ." What is the student's error?

A.

The student substituted the input $n = 49$ in place of the previous output $g(n-1)$

B.

The student forgot to include the initial value $g(0) = 2$

C.

The rule is multiplication, not addition; the student should have written $g(50) = g(49) \cdot 3$

D.

The student should use an explicit formula — recursion cannot be used for large values of n

5.

For $f(0) = 1$ and $f(n) = f(n-1) + 4$ , compute $f(3)$ using the recursive definition. Verify with the explicit formula $f(n) = 4n + 1$ . Enter the value of $f(3)$ .

D

Word Problems

1.

A plumber charges a $75 base fee plus $40 per hour.

The explicit formula for total cost is $C(h) = 40h + 75$ . Find the cost for 8 hours of work.

2.

A rabbit population is modeled recursively: $P(0) = 50$ and $P(n) = 2 \cdot P(n-1) - 10$ , where $n$ is the generation number.

What is the population in generation 3?

3.

A phone plan charges $30 for up to 2 GB of data, then $10 for each additional GB over 2 GB, plus $0.05 per text message.

Write a numbered procedural description for computing the monthly cost given data (in GB) and number of texts. Explain in one sentence why this is a valid function definition.

E

Error Analysis

1.

A student is given $f(0) = 8$ and $f(n) = f(n-1) + 6$ . To find $f(100)$ , the student writes: " $f(100) = f(99) + 6 = 99 + 6 = 105$ ."

What error did the student make?

A.

The student substituted the input value $n = 99$ in place of the output $f(99)$

B.

The student forgot the initial value $f(0) = 8$

C.

Recursion cannot evaluate $f(100)$ — an explicit formula is required

D.

The student used the wrong common difference — it should be 5, not 6

2.

A student writes a recursive definition for the sequence 3, 8, 13, 18, ...: " $f(n) = f(n-1) + 5$ "

What is missing from this recursive definition, and why does it matter?

A.

The common difference is wrong — it should be 4

B.

The initial value is missing — without $f(1) = 3$ (or $f(0) = 3$ ), infinitely many different sequences satisfy this rule

C.

Only the explicit formula is needed — recursive definitions are optional

D.

Nothing is missing — the rule fully defines the sequence

F

Challenge / Extension

1.

A recursive definition is: $g(0) = 200$ and $g(n) = g(n-1) \cdot 0.8$ . Translate to explicit form: $g(n) = \text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}} \cdot (\text{\hspace{0.2em}\fbox{\phantom{000000}}\hspace{0.2em}})^n$ . Then find $g(5)$ rounded to the nearest whole number: ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

initial value: