Identify Functions
What is a function?
A type of relation where each input value has exactly one output value.
A car can be rented for a flat fee plus an additional amount per day. The function f(x) = 55 + 50x models the total cost of the car in dollars, f(x), after x days.
What does 55 represent?
The flat fee for the car rental.
Evaluate f(x) = -5x at x = -1.
f(-1) = -5(-1)
f(-1) = 5
Select the correct description of the following pattern: 2, 5, 10, 17,...
A. Each term is twice the previous term plus one.
B. Consecutive odd numbers are being added to each term.
C. Each term increases by three times the previous term minus one.
B. Consecutive odd numbers are being added to each term.
What is the constant value that is added to or subtracted from each term in an arithmetic sequence to determine the next term?
common difference
Is this data set a function? Why or why not?
{(2, 3), (0, 0), (5, 3), (8, –1)}
Yes, this is a function because every input (2, 0, 5, 8) has only one output.
Kevyn has $50 saved in his piggy bank right now, and each week he wants to save an additional amount of money. The total amount in his piggy bank can be modeled by the equation y = 10x + 50, where x is the number of weeks.
How much is Kevyn adding to his piggy bank weekly?
He is adding $10 weekly.
Evaluate the following function at x = 4.
f(x) = 2x2 – 1
f(4) = 2(4)2 – 1
= 2(16) – 1
= 32 – 1
= 31
f(4) = 31
2, 2, 4, 12, 16, 80, 86, …
Describe the pattern in the sequence
Multiply by 1, add 2, multiply by 3, add 4, multiply by 5, add 6, etc.
Determine if the sequence is arithmetic, geometric, or neither. Explain.
19, 15, 11, 7, ...
Arithmetic because a constant of 4 is being subtracted from each term.
What are the domain and range of a function?
The domain is the set of all inputs (or x-values).
The range is the set of all outputs (y-values or f(x) values).
A shop sells different flavors of popcorn by the pound. They charge $7 per pound plus an additional $3 for the tin to hold the popcorn.
Write a function that models this scenario.
f(x) = 7x + 3
Evaluate f(x) = -x2 – 2x + 7 at x = -2.
f(-2) = -(-2)2 – 2(-2) + 7
f(-2) = -4 + 4 + 7
f(-2) = 7
Geoffrey says the following pattern is both arithmetic and geometric because the rule is add 3, then multiply by -1.
1, 4, -4, -1, 1, 4, -4, -1, 1, …
Do you agree with him? Explain your reasoning.
No. Though the rule involves both adding and multiplying, there is no common difference or common ratio, so it is neither arithmetic nor geometric.
Consider the sequence: 1, 2, 4, 7, 11, ...
Is it arithmetic or geometric? Why?
Neither because there is no common ratio or common difference.
Given the domain x = 2, -5, and 0, identify the range of the following function:
f(x) = -3x – 5
f(2) = -3(2) - 5 = -6 - 5 = -11
f(-5) = -3(-5) - 5 = 15 - 5 = 10
f(0) = -3(0) - 5 = 0 - 5 = -5
range: {-11, 10, -5}
Abdullah is studying the growth process of daffodils. He plants 3 daffodils in his garden, and observes each week that the number of daffodils in his garden doubles.
Write a function for the number of daffodils in Abdullah’s garden after x weeks.
y = 3(2)x
Evaluate g(4m) for the function g(x) = 5 – 2x.
g(4m) = 5 – 2(4m)
g(4m) = 5 – 8m
Create a sequence that is neither arithmetic nor geometric. Explain the rule for your sequence.
(Answers will vary.)
Describe the rule, then find the next three terms in the sequence.
3, 6, 5, 10, 9, 18, 17,...
Rule: multiply by 2, then subtract 1
Next three terms: 34, 33, 66
A bug travels up a tree, from the ground, over a 30-second interval. It travels fast at first, then stops for 10 seconds, then proceeds slowly, speeding up as it goes. Does this scenario represent a linear or non-linear function? Sketch it!
non-linear (Sketches will vary.)
The 44th president has put out the first pair of official presidential sneakers – the Barry Zeros. The value of a pair of Barry Zeros depreciates by 5% every year, t, that they are worn. Your new pair of Barry Zeros costs $300.
Write an equation that represents how much your pair will be worth after 9 years of wear.
C = $300(1 – 0.05)9 = $300(0.95)9= $189.07
Write 2 different functions such that f(3) = 2.
(Answers will vary.) Sample responses:
f(x) = x – 1
f(x) = 2x – 4
f(x) = 3x – 7
A pattern shows shapes formed by an increasing number of circles. The first figure has 1 circle, the second figure has 6 circles, the third figure has 15 circles, and the fourth figure has 28 circles.
Write a rule to determine how many circles are in a figure.
1, 6, 15, 28, …
an = 2n2 – n, OR
an = n(2n – 1), OR
an = n2 + n(n – 1)
14, 10, 6, 2, -2, ...
Write a recursive formula for the sequence above.
an = a(n-1) – 4 when n ≥ 1, a1 = 14