Functions- Unit 8

What is a function?

A relation that assigns exactly 1 y value for each X value.

What is a relation?

A set of ordered pairs (somehow, they are related) (5,8) (7,2)

What would it look like for f(x)= -4x - 7 F(-3) + 1

Because the +1 is *behind* the f(x) part, the +1 goes after the f(X) equation in brackets once you plug in the -3 for X. [-4 (-3) + 7] +1

If you have a graph of a person's height while jumping of a diving board, where would the graph begin?

Because the person is already on the board, the height should start a little bit above zero.

When it asks what is happening between a and b minutes, and it is plateauing:

Format: the water level remains unchanged between a and b minutes.

How do you tell if a group of coordinates is a function?

If each X value only has one individual y value, then it is a function. If the same X value is assigned to different y values, it is *not* a function!

Would the graph of a basketball be constant or inconsistent?

Inconsistent!! (CURVY LINES!)

Is a graph of a circle a function?

No!

What is the format you write for range?

R: {y I y < 5 } (Value is an example)

Note:

They will usually go in a line like the inequalities we graphed with the points meeting. But they don't have to!

Remember, if it says F(t) is the number of people that own cell phones (for example), *in millions*, how do you write your answer?

You write your answer as you answer with the word millions after it.

What is an example of an equation that is a function?

y= 1/3x + 2

Note:

Don't forget to label the different sections of the graph! And explain, if you need to.

You will most likely be asked to graph problems before determining if they are a function or not. Should you connect the coordinates on these graphs?

No.

If the circle is *NOT* closed in on one of the points that marks the beginning or end of domain/range values, it would just be < or > because it is * NOT exactly equal to*

Note:

P(t) is the population of a country t years after 2000. Interpret the following. P(13) - P(12) = 10

The population between the years 2012 and 2013 increased by 10 people. (Positive number)

What is the output?

The value that is the outcome of an equation (usually y)

How do you determine which factor is X and which factor is y in a relationship presented In a table?

X *always* comes before y!

Domain

X values

Is y= I x - 2 I + 3 a function?

Yes

When looking at qualitative relationship graphs and determining what situation it accords with, remember to:

-Pay attention to the starting point -Think about *how long the duration of time is* and what it would look like as a whole. -Notice if the change is steady or varying. -Pay attention to whether or not the graph ever ends and goes back down to 0.

Find the *domain* of the problem given the range. y= 4x + 5 {5, 10, 12}

5= 4x + 5 X= 0 10=4x + 5 X= 1 1/4 12=4x + 5 X= 7/4 Answer= {0, 1 1/4, 7/4}

When it asks when does the water level reach 140 cm, what do you have to remember?

Be careful because, remember it could also reach that somewhere along the line even if it's not an exact point. So it could be at like 7 months or approximately 8.5 months.

P(t) is the population of a country t years after 2000. Interpret the following. P(13) - P(12) = -10

Between the years 2012 and 2013, the population of the country decrease by 10 people. (Negative number)

Remember, if you're measuring something on time, and you do something for 2 *minutes*, the line should be much shorter than the line for if you did something for 1 *hour!*

Check that on your problems!

What's the difference between a constant rate and different (inconstant rate)

Constant rate= linear graphs; straight line

If the domain and range values *are* infinite, how do you write them?

D: { ì } R: { ì } (Infinity sign looks like i or upside down exclamation point)

What is the format you write for domain?

D: {x I x < 5} (Value is an Example)

What should the graph do if something stops?

Either go back down to 0 (if you're measuring speed) or plateau (if you're measuring distance)

Graph height of baby up to 2 years old.

Graph should be wavy and in a sort of pattern. The baby will not grow all at once, he will grow a little bit (increase), then stay that height (constant), then grow a little more rapidly, then stay that height, etc. Lind should just taper off or end where it is at the end.

How do you determine if a graph is a function by using the vertical line test?

Hold your pencil vertically over the graph. If the line of the graph is perpendicular to the pencil more than once at any point on the graph (it crosses/ runs into the pencil twice or more), the graph is not a function. If it only crosses once, it is a function. *(There can't be more than one coordinate on each vertical line)*

When you must interpret the graph and write a scenario:

Label each part of the graph using a), b), and c) and then using the key, explain each part. Remember to pay attention to what each part of the graph is doing and where it's at on the *y-axis! (Time)* As long as what you're saying makes sense and is correct in terms of the graph, it should be fine.

Are these relations a function? (-6,2) (8,9) (3, -2) (-6,7)

No, the relations are not a function because the X value of -6 is assigned to 2 separate y values.

What do you do when f(x)= 2x - 3 and f(1)?

Since they layout if f(x), they're basically saying 1 is equal to x, so you plug in 1 for x. Solve and then format your final answer as f(x)= answer/solution.

Find the *range* of the problem given the domain. How do you do this?

Since, they give you *the domain*, and the domain is the x values, you will plug each of the domain values into x's place in the equation and solve for y. Once you have found the Y value of the equation for *all three* x values, list the Y values as your answer in brackets *in order from least to greatest.*

Find the *domain* of the problem given the range. How do you do this?

Since, they give you *the range*, and the range is the y values, you will plug each of the range values into y's place in the equation and solve for x. Once you have found the X value of the equation for *all three* y values, list the X values for your answer in brackets *in order from least to greatest.*

What would the graph of an oven look like?

Slowly rising (heating up), small plateau (reaches set temperature), then goes down as the heat shuts off, then comes up (heat turns back on when it gets too cold), then cools/goes down, and so on. There should be a steady change of temps with the fan until it turns off.

Graph height of a basketball (if it says just ball, assume the same thing) that is dropped from a height of 30 feet.

The ball is dropped from a height of 30 feet and its height decreases at a constant rate until it hits the ground. The ball bounces so the height increases and then decreases (at a constant rate) until it gradually comes to a stop, in which the height would finally be 0 feet.

If you are graphing your your distance from the library when you are driving halfway to the library and then the speed when you decide to walk the rest of the way. What will the difference look like on the graph?

The first will be somewhat fast and inlinjng steeper than the first because you are getting closer to the library. But then, because your speed decreases and you aren't getting to the library as fast as when you're driving, the line will gradually increase, but at a much more gradual rate.

If a table is set up vertically, which factor is the left?The right?

The left is the X and the right is the y! So remember, *with vertical, it's always:* X= left Y= right

Quantitative Relationships: P(t) is the population of a country t years after 2000. Interpret the following. P(0) = 750,000

The number of people in a country in 2000 is 750,000. (Because the t represents years *since or after* 2000, and it says 0 years after 2000, it's just going to be 2000.

If his distance is decreasing at a constant rate, what does that mean is happening to the speed?

The speed of the person is increasing at a constant rate as he quickly pedals back to his car.

What is the input?

The value you put in an equation (Usually x)

Range

Y values

When it asks what is happening from 15-20 minutes:

(Remember to use complete sentences when answering these!) Format like this: The height of the water is decreasing/increasing at a constant rate.

When it asks is the water level (for example) decreasing at a faster rate between a and b minutes or c and d minutes how would you justify your answer?

A and b minutes (for example) because there is a steeper steeper slope.

What's the difference between a constant rate and different (inconstant rate)

Different rate= wavy graph and wavy line

What is the difference between domain and range values?

Domain usually refers to X values and range usually refers to y values.

If it asks if a graph is a function by determining whether or not it passes the vertical line test, what do you say?

Ex: No, the graph is not a function because it does not pass the vertical line test. Using/ holding the pencil vertically at various points on the line, the line of the graph intersects the pencil horizontally in more than one place at a time, meaning the graph is not a function because the X value (of -5 for example, if you *can* determine it) is assigned to two/multiple different y values.

If graph has numbers on the side, what do you do?

Incorporate the numbers in your description! Example: the speed of the car increases at a constant rate until it reaches 50 mph and remains at that speed.

What is the difference between independent and dependent variables?

Independent variables are values that can be freely chosen and and are the values that are inputted. They directly affect the dependent variable and are usually the X values. Dependent variables depend on the input of the independent variables and are completely dependent on them. They are usually the y values.

How do you state the domain and range of a graph?

Look at the graph. 1) First of all, are there arrows on the end of the graph? If there are arrows, that means the domain and range can be infinite, *but only if the line and arrows extend all across the graph and not just infinite on a single value.* 2) If it's not infinite, look at what values the domain ranges from and if they are greater than one number and less than the other. For example, say the line of the graph has domain values from -4 to 7. 3) Look at -4. Since y is greater than -4 (It has to since it extends to positive 7) or equal to -4 (it has point/s on that direct domain value) and is less than or equal to 7 for the same reasons, you would write the domain value as the following: D: {x I -4 ≤ x ≤ 7} 4) Do the same with the y/range values.

Remember to pay attention to the arrows on the graph and if it's actually infinity values or not. For example, if the end of the line of the graph went down to the y value of -4 and continued across that horizontal line with an arrow showing you it continued on, and it did the same thing on the other side, the domain values would/could (depending on the situation) be infinite, but not the y!

Remember, just because it has arrows at the end of the graph does *not* mean everything necessarily has infinite values.

When graphing the amount of rainfall on a spring day: It storms violently in the afternoon then tapers off to a sprinkle.

Remember, on problems like this, just because it slows down to a sprinkle, that doesn't mean rain has *stopped* collecting. *It is still rajning*, just not *as hard.* So it would still go up, just at a more gradual rate.

If you are graphing your your speed from the library when you are driving halfway to the library and then the speed when you decide to walk the rest of the way. What will the difference look like on the graph?

The difference in the graph from the previous question is that once you decline in speed, since you're my measuring distance, you're measuring speed, your speed on the graph would decrease gradually.

When you are measuring distance, and the graph has plateaued, what does that mean for the speed of the person?

The person has stopped and is not moving.

P(t) is the population of a country t years after 2000. Interpret the following. P(10) = 1,250,000

The population in 2010 is 1,250,000 people. (Add 10 years to 2000)

C(t) is is temperature of the sweet potato after t minutes in the oven. Interpret the following: C(15) < C (23)

The temperature of the potato after it has been in the oven for 15 minutes is less than the temperature of that at 23 minutes.

C(t) is is temperature of the sweet potato after t minutes in the oven. Interpret the following: C(0)= 65

The temperature of the potato when you placed it in the oven was 65 degrees.

C(t) is is temperature of the sweet potato after t minutes in the oven. *You take it out after 45 minutes.*Interpret the following: C(45) > C (50)

The temperature of the sweet potato at 45 minutes is greater than that at 60 minutes. This makes sense because you took the sweet potato out of the oven after 45 minutes, and the temperature would still be hot. But after 60 minutes, it is had time to cool, and that is why the temperature is less at 65 minutes.

If a table is set up horizontally, which factor is the top? The bottom?

The top will be the X and the bottom will be.l the y! With horizontal, *always remember:* X= top Y= bottom

Would graph of heart rate or height of person be wavy or straight? (ACCORDING TO THE GRAPH)

Wavy/inconsistent

Independent variable

X axis A variable in an equation that's input may be freely chosen, and it directly affects the dependent variable.

Dependent variable

Y axis A variable whose value depends on the input of the independent variable, and is completely *dependent* on it.

Find the *range* of the problem given the domain. y= 4x + 5 {5, 10, 12}

Y= 4(5) + 5 Y= 25 y= 4(10) + 5 Y= 45 Y= 4(12) * 5 Y= 53 Answer(Y=): {25,45,53}

Are these relations a function? (-4,6) (0,5)(3,2) (1,9) (0,5)

Yes, because the X value 0 is only assigned to 1 y value. (5 is the same number)

Are these relations a function? (9,7) (-10,7) (1,4) (0,-8)

Yes, the relations are a function because each X value is assigned to only *one* individual y value.

How should you write your list your numbers when your solving for the domain/range given the range/domain in these type of problems?

You should always list your numbers in order *from least to greatest!* Also, your values should be in *a bracket!*

What would you do for the following: G(X)= .... G(-4t)

You would just plug -4t in for X and solve as much as you can. Obviously, you won't be able to solve because there is an unknown variable, so after you solve as much as you can, you just leave it like that. It's okay to do that!

What would you do for the following: F(x)= 2x - 3 6 - f(5)

You would still substitute the 5 in for X. However, there is a subtraction operation as well. Because the ( 6 - ...)is *in front* of the f(x/5), you put the 6 - *in front of the 2x - 3! *However, you need to do the 2x - 3 first, so that part would be in brackets and the 6 - would be outside of the brackets and in front of the brackets. So it would look like: 6- [2(5) - 3] 6-[10- 3] 6 - (7) F(X)= -1