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Math 3 Module 1: Functions & Their Inverses

1.1 Ready, Set, Go!

Question 1
To "undo" multiplication, you need to use the opposite operation - division! The process of "undoing" is known as "inverse operations". This video shows you an example of using inverse operations to isolate the variable like you need to do for this problem: https://www.youtube.com/watch?v=Velg__aLrmo

Question 2
To "undo" division, you need to use the opposite operation - multiplication! The process of "undoing" is known as "inverse operations". This video shows you an example of using inverse operations to isolate the variable like you need to do for this problem: https://www.youtube.com/watch?v=Velg__aLrmo

Question 3
To "undo" addition, you need to use the opposite operation - subtraction! The process of "undoing" is known as "inverse operations". This video shows you a similar example where you need to use inverse operations to isolate the variable: https://www.youtube.com/watch?v=QQoPAU5L52s

Question 4
How do you undo a square root? Check out this example: https://www.youtube.com/watch?v=IfnLx-8-zD8

Question 5
This is a multistep problem! This video walks through a similar example with a cube root: https://www.youtube.com/watch?v=tXyZsQp2YJg

Question 6
In this problem, x is raised to the power of 4. To undo that, you need to perform the inverse operation, like this: https://www.youtube.com/watch?v=JhMlTDx2rP8

Question 7
For this question, the entire term is squared (or raised to the power of 2). To undo the power and solve for x, check out this video: https://www.youtube.com/watch?v=JhMlTDx2rP8

Question 8

You're given a table and need to convert some of the data in the table to ordered pairs where cats are "inputs" and dogs are "outputs". In this case, an input is the value of x and the output is the value of y.

Question 9
Using the data in the table, you have to come up with a function f(c) = d that relates the number of cats (the input) to the number of dogs (the output). This video shows you how to write out a function from data in a table: https://www.youtube.com/watch?v=hoJ6Q1psS2k

Question 10
Like question 8, you need to convert some of the data in the table to ordered pairs but now dogs are "inputs" and cats are "outputs".

Question
 11
This question is very similar to question 9, except now you need to write out the function f(d) = c that takes dogs as the input and cats as the output! This video shows you how to write out a function from data in a table: https://www.youtube.com/watch?v=hoJ6Q1psS2k

Questions 12
All ordered pairs have the format (x, y) but in question 8, x was cats and y was dogs and in question 10, x was dogs and y was cats.

Questions 13
You can think of the two functions as opposites - one goes from cats to dogs and the other goes from dogs to cats. The mathematical term we use to describe them is "inverse functions".

Questions 14​
To calculate f(10), set x = 10 in the function f(x) = x.

Questions 15​
To calculate f(-2), set x = -2 in the function f(x) = x.

Questions 16

To calculate f(a), set x = a in the function f(x) = x. We don't what the value of a is, so it will appear in your final answer.

Question 17

To calculate f(a + b), set x = a + b in the function f(x) = x. We don't what the values of a or b are, so they'll appear in your final answer.

Questions 18
To calculate g(10), set x = 10 in the function g(x) = 5x - 12.

Questions 19
To calculate g(-2), set x = -2 in the function g(x) = 5x - 12.

Questions 20
To calculate g(a), set x = a in the function g(x) = 5x - 12. We don't what the value of a is, so it will appear in your final answer.

Questions 21
To calculate g(a + b), set x = a +b in the function g(x) = 5x - 12. We don’t know what the values of a or b are, so they'll appear in your final answer.

Questions 22​
To calculate h(10), set x = 10 in the function h(x) = x^2 + 4x - 7.

Questions 23

To calculate h(-2), set x = -2 in the function h(x) = x^2 + 4x - 7.

Questions 24
To calculate h(a), set x = a in the function h(x) = x^2 + 4x - 7. We don't what the value of a is, so it will appear in your final answer.

Questions 25
To calculate h(a + b), set x = a + b in the function h(x) = x^2 + 4x - 7. We don't what the values of a or b are, so they'll appear in your final answer.

1.2 Ready, Set, Go!

Questions 1 - 8

In this question you need to evaluate the function. This video gives you some similar examples: https://www.youtube.com/watch?v=pJjcqlGUPg4

Questions 9 - 12

You need to figure out what the pattern is and then add the missing entries, like this video shows: https://www.youtube.com/watch?v=1Ww6U__uy6A

Question 13

All you need to do is draw the next figures in the pattern. This video shows a similar pattern: https://www.youtube.com/watch?v=No4VLQLrdr0

Question 14

Look at expression (a), (b) and (c) and decide which one fits the scenario.

Question 15

If "n" is the number of tiles and "s" is the number of steps, then number of tiles "n" is equal to 3 times "s" minus 2.

Question 16

Add up the numbers in the right column in the table to start with. Then the easiest way to do this is to set s = 1, 2, 3, ... and calculate n for each expression and then compare it to the values in the table and see which expression matches.

I also noticed that one of the expressions had a formula for odd numbers in it (2s - 1) and then adds the number of tiles above the base which is 0 when s=1, and is 1 when s=2.

Questions 17 - 24

You need to turn this multiplication expression into an exponent expression. This video reviews exponents: https://www.youtube.com/watch?v=XZRQhkii0h0

Questions 25 - 28

In this problem you are multiplying an exponent by a whole number. This video provides similar examples: https://www.youtube.com/watch?v=BVwFp0-pmRE

1.3 Ready, Set, Go!

Question 1

For part (A) look along the table for f(n) = 12 and find the value for n above the 12. 

 

For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

Question 2

For part (A) look along the table for f(n) = 17 and find the value for n above the 17.

For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

Question 3

For part (A) look along the table for f(n) = 32 and find the value for n above the 32.

For part (B), you need to evaluate f(n - 1). Use the value that you found for n in part (A). For example, if you found n = 101, then f(n - 1) would be f(100) (which is outside of the table)

Question 4

For part (A) look along the table for f(n) = 2 and find the value for n above the 2. 

 

For part (B), you need to evaluate f(n + 3). Use the value that you found for n in part (A). For example, if you found n = 3, then f(n + 3) would be f(3+3) or f(6). Which means, when n is 6, what is f(n)?

Question 5

For part (A) look along the table for f(n) = 27 and find the value for n above the 27. 

 

For part (B), you need to evaluate f(n - 6). Use the value that you found for n in part (A). For example, if you found n = 8, then f(n - 6) would be f(8-6) or f(2). Which means, when n is 2, what is f(n)?

Question 6

For part (A) look along the table for f(n) = -8 and find the value for n above the -8. 

 

For part (B), you need to evaluate f(n + 9). Use the value that you found for n in part (A). For example, if you found n = 1, then f(n + 9) would be f(1+9) or f(10). Which means, when n is 10, what is f(n)?

Question 7

Set x = 4 in the explicit equation and evaluate to get the value of the 4th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Question 8

Set x = 50 in the explicit equation and evaluate to get the value of the 50th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Question 9

Set x = 20 and evaluate the explicit equation to find the value of the 20th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Question 10

Set x = 9 and evaluate the explicit equation to find the value of the 20th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Question 11

Set x = 5 and evaluate the explicit equation to find the value of the 20th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Question 12

Set x = 7 and evaluate the explicit equation to find the value of the 20th term.

If you want to use the recursive formula, then this video is helpful: https://www.youtube.com/watch?v=RjsyEWDEQe0

Questions 13 - 17

In this question you have to evaluate the exponential and fill the table, like the example in this video: https://www.youtube.com/watch?v=PEtIQqvIoGU

1.4 Ready, Set, Go!

Questions 1 - 8

In this question you need to evaluate the function at the value you are given. This video gives you some similar examples: https://www.youtube.com/watch?v=pJjcqlGUPg4

Questions 9 - 12

For part (A) and (B) you need to figure out the pattern and finish the sequence, like in this video: https://www.youtube.com/watch?v=EadlY9iuIIM

In part (C) you need to give the recursive and explicit functions that describes the sequence, like this: https://www.youtube.com/watch?v=1W2Ziv4m0Bo (IMPORTANT NOTE: in this video, they use f(n) to describe "now" and f(n-1) to describe "previous")

Questions 13 - 14

If you need help reading coordinates off graphs, this video might be a good refresher: https://www.youtube.com/watch?v=j6LGxJhc8Kk

Question 15

Look at Olaf's original elevation at noon and subtract his elevation at 2pm to find the distance has has walked down the mountain. If you need help reading coordinates off graphs, this video might be a good refresher: https://www.youtube.com/watch?v=j6LGxJhc8Kk

Question 16

Functions such as f(n) usually have inputs such as time and distance. Based on the graph you're given, the function is probably calculating the elevation at time n.

1.5 Ready, Set, Go!

Questions 1 - 4

To find the rate of change in a table, check out this video: https://www.youtube.com/watch?v=wasEKQkA7MA

For help finding the rate of change on a graph, take a look at this other video: https://www.youtube.com/watch?v=MYCH7gswI4k

Questions 5 - 6

This video explains how to find the explicit and recursive formulas for geometric sequences: https://www.youtube.com/watch?v=8a1a5A3CfdQ

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

Question 7

Use the information given to create a table with 4 rows (like the table in question 6) and fill the right column with the amount of money left in Claire's account each month. After 1 month, she takes out half the money and is left with $150. After 2 months, she takes out half of what was in there from the previous month. Calculate how much she will have after 2, 3, and 4 months to fill out the table. From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

Question 8

Use the information given to create a table with 4 rows (like the table in question 6) and fill out the right column with the number of people that will receive the chain letter each time. For example, on day 1 Tania sends the letter to 4 friends, so 4 people have received it. On day 2, those 4 friends send the letter to another 4 friends each, so 16 people receive the letter that day. Figure out how many people receive letters on days 3 and 4! From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

Question 9

Count up the number of dots in each sequential figure and put it into a table. From there, you can find the recursive and explicit formulas for geometric sequences like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ.

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

Questions 10 - 13

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

 

This one goes through finding the explicit formula: https://www.youtube.com/watch?v=T1ZBizIL5Lw

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

Question 14

Count up the number of dots in each sequential figure and put it into a table. From there, you can find the recursive and explicit formulas like this: https://www.youtube.com/watch?v=8a1a5A3CfdQ

If you want a refresher on graphing, this video shows you what you can expect arithmetic and geometric sequences to look like when they are graphed: https://www.youtube.com/watch?v=jU6JTdhHVuI

1.6 Ready, Set, Go!

Questions 1 - 6

This video shows you how to find the missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=g7Tl3clSDYw

Questions 7 - 9

I found a video that shows you how to find the recursive formula of an arithmetic sequence given two terms. The example uses different notation but don't let that scare you! His a₃ is the same as f(3)). https://www.youtube.com/watch?v=VguyPrGaXLk

Questions 10 - 12

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 13 - 16

This question asks you to evaluate the exponential function they've given you. This video shows you how to evaluate exponential functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

Question 17

Look at the parentheses in question 15, how do they change the effect of the exponent? https://www.youtube.com/watch?v=pJjcqlGUPg4

Questions 18 - 19

These questions asks you to evaluate the linear function they've given you. This video shows you how to evaluate linear functions: https://www.youtube.com/watch?v=GQIcZeKR9RI

1.7 Ready, Set, Go!

Questions 1 - 4

Look at the pattern and fill in the missing numbers. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

Questions 5 - 8

Look at the pattern to decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

Question 9

Use the story to write out the first 4 or 5 numbers in the sequence. Look at the pattern to decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 10 - 12

This explains the difference between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 13  - 18

This video shows you how to find slope of the line that runs through two points: https://www.youtube.com/watch?v=8trWFtwyUMU

If you need a refresher on plotting coordinates on a graph, here's a good video for that: https://www.youtube.com/watch?v=s7NKLWXkEEE

1.8 Ready, Set, Go!

Questions 1 - 4

This video shows you how to find the common ratio for a geometric sequence: https://youtu.be/-rofZ1OtTik

Questions 5 - 8

Start by looking at the pattern and decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

 

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

Questions 9 - 12

You need to find the equation of the line on the graph. This video has an example of a line with a positive slope: https://www.youtube.com/watch?v=mmWf_oLTNSQ

 

And this video is an example of a line with a negative slope: https://www.youtube.com/watch?v=dBFbFMvEwQg

1.9 Ready, Set, Go!

Questions 1 - 2

This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

Questions 3 - 6

This video has examples of finding missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=g7Tl3clSDYw

This video shows you how to find the formula for the n-th term in an arithmetic sequence:

https://www.youtube.com/watch?v=lj_X9JVSF8k

Questions 7 - 11

Start by looking at the pattern and decide if the sequence is arithmetic or geometric. This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

 

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

To find the explicit form of the geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

 

When you have explicit formula, you can convert it to a recursive formula like this: https://www.youtube.com/watch?v=Iq7a2vEsT-o

1.10 Ready, Set, Go!

Questions 1 - 5

You are given the first term and common difference, d, for an arithmetic sequence f(d). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=eR_kQf-1fks

You are given the first term and common ratio, r, for a geometric sequence g(r). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=3uiTQ6JSMC0

 

Question 6

Arithmetic sequences are straight lines on a graph and geometric sequences are exponential curves. This video compares linear and exponential curves - you can decide if exponential curves always outgrow lines! https://www.youtube.com/watch?v=TVX9pSPJwYA

Questions 7 - 10

This video has examples of finding missing terms in an arithmetic sequence: https://www.youtube.com/watch?v=tqli6fOnph8

This video shows you how to find the formula for the n-th term in an geometric sequence:

https://www.youtube.com/watch?v=IGFQXInm-co

Question 11

You are given the first term and common ratio, r, for an geometric sequence g(r). You need to find the first 5 terms of the sequence with that information. This video shows you how to do a similar question - notice that you already have the common difference, so you don't need to calculate it: https://www.youtube.com/watch?v=3uiTQ6JSMC0

Questions 12 - 13

To find the explicit formula for a geometric sequence, check out this video: https://www.youtube.com/watch?v=faBM7b8KKjM

Question 14

Plug the value x = 100 into the explicit expressions you found in questions 11., 12., and 13. and see which value is largest!

1.11 Ready, Set, Go!

Question 1

This video explains some of the differences and similarities between arithmetic and geometric sequences: https://www.youtube.com/watch?v=9aAWRRzGTXI

 

Khan Academy has a good explanation of arithmetic sequences: https://www.youtube.com/watch?v=_cooC3yG_p0

 

Question 2

"Create a context" means "come up with a story that this sequence would apply to. You can come up with any story that fits the data!

This video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

 

This one goes through finding the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

If you need a refresher on plotting coordinates on a graph, here's a good video for that: https://www.youtube.com/watch?v=s7NKLWXkEEE

Question 3

In this question you are given the recursive equation for the sequence. I would use this formula to list out the first 4 or 5 points in the table like this: https://www.youtube.com/watch?v=IFHZQ6MaG6w

When you've filled the table, this video shows you how to find the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

Question 4

In this question you are given the explicit equation for the sequence. Evaluate the function f(n) when n = 1, n = 2, ..., n = 5.

 

When you've filled the table, this video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

Question 5

When you've filled the table, this video shows you how to find the recursive formula for arithmetic sequences: https://www.youtube.com/watch?v=ViLt2WI0XSg

Questions 6 - 8

In this question you are given the recursive equation for the sequence. I would use this formula to list out the first 4 or 5 points in the table like this: https://www.youtube.com/watch?v=IFHZQ6MaG6w

When you've filled the table, this video shows you how to find the explicit formula for arithmetic sequences: https://www.youtube.com/watch?v=T1ZBizIL5Lw

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