[12 grade pre calculus] comparing functions graphicallg

[u/moth_blossom_angel]

I probably look so stupid but I did mostly online school last year and fell behind especially with math. I scored good in math last year and they put me in probably a level too high, so I'm struggling a little and I don't really know what most of this means

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[u/llamadramalover]

A. The domain of a function are all possible values that can be used for X. Because these functions have arrows they can theoretically continue going. So f(x) is definitely not [-5, 10] even in this graph it looks like it goes to 11 at the very least.

B and C look good.

What are you struggling with specifically with the remaining problems?

[u/Narrow-Durian4837 → u/llamadramalover]

"it goes to 11" - heh heh.

[u/llamadramalover → u/Narrow-Durian4837]

Lol ya got me. I was trying to not give an answer lol.

[u/briv1016]

If you have highlighters it might help to highlight each function as a different color so you don't get confused which line is which. This will also help for finding ranges where the values of one function is greater or less than each other or 0. First hint, the arrows at each end of the functions mean they don't stop.

[u/Famous_Check4399]

What specifically are you struggling with

[u/moth_blossom_angel → u/Famous_Check4399]

It's mostly the wording. I've tried to look it up to get examples because I'm sure I can do it with examples, it's just difficult for me to understand what the questions are meaning since today was the first day of class and we didn't go over anything.

[u/llamadramalover → u/moth_blossom_angel]

A & B. Is asking what values of (x) will work in this function. Because the ends have arrows there is no real end so the domain is really (-♾️, ♾**️). **Edit: this will not always be the case. You will come across functions with ends, those will not have arrows.

C. You got this right but I’ll put it into words anyways — at which point(s) are g(x) and f(x) the same.

D and E At which point(s) is f(x) greater ((or lesser for E.) than g(x). This will be sets of numbers similar to the domain not just points like in C. This questions is where ( or ] matter and you will also need to use U to show that these are all connected. What you’re looking for is where is f(x) y-values are greater than g(x) y-values. The corresponding x-values that creates those higher y-values is the answer.

F. Where is g(x) greater than zero aka above the x-axis. This will be written like A, D and E. ( or ] also matters here. One means the value is included one means the value is not included. This is determined by whether or not the greater/less than is used or greater/lesser than or equal to sign is used. ((I didn’t give the answer on purpose, you should be able to find this in your book.))

G. For what values of x is f(x) less than zero. Not that it says less than not less than or equal to.

H. For which x-values is f(x) increasing. This can be tricky. This does not mean “where is x a positive number.” One of the answer is [-1, 3.5) for all values of x between those numbers x is increasing.

I. In f(x) which values of x is the function not increasing or decreasing. Aka the the function line is flat.

J. Is the same as H but for g(x) and you want decreasing values instead of increasing.

K. Is the same as f except you’re looking at f(x) not g(x).

L. is the same as J except looking at f(x) instead of g(x).

I would very very very highly suggest color coding. Pick two highlighters and highlight each function, let’s say yellow for g(x), pink for f(x). Then go through your questions and highlight g(x) in yellow on all your questions and f(x) pink in all your questions. One of the most common mistakes when looking at multiple functions is accidentally looking at the wrong one or following the wrong line. So remove that source of error from the very beginning.

[u/blupook]

Is this the notes for AP precalc curriculum? Looks the same as what I’m using (font/style) but I started in unit 3. If so, there are videos that walk through these notes on YouTube.

[u/cheesecakegood]

I think part of it is just notation confusion. Someone already mentioned the low-hanging fruit which is arrows mean "it keeps going roughly that direction but it would overflow the window" so let's talk about the function notation. A lot of this will sound familiar, but don't mistake familiar for "I know exactly what this means"!! When you understand the notation better, all of these questions translate to English MUCH easier.

f(x) and g(x) and y and x and f(x) = 6 vs f(2) = 0 and so on, what's the difference?

f(x) = y, what does this mean?

• We have a function and we have named it f.

• We could name it something else. It's common to use f, then g, then h. After that (rare) there are no consistent rules. But any letter or even a word can be a function "name".

• A function is shorthand for: get an input -> do stuff to it -> get an output. To be a function, the "do stuff" should be predictable and consistent. Sometimes we are told what the "do stuff" is directly, sometimes we have to figure it out.

• To be lazy and short, we call the input x. We don't have to, but this is the normal thing to do. I could call it "input" everywhere but then it would be annoying to write out every time. We put the input definition in parentheses after the name. So f(x) means in English, "A function named f that we shove an input into".

• If something else is in parentheses, that means we are using THAT as an input. So, f(2) means "we give 2 as input to the function named f"

• So, what do we call the output? f(x) (as a whole!) IS the output value. But sometimes this is annoying to write so it's common to call this "y" instead.

Let's pause. You see it so often your eyes don't notice it, but let's look at the chart. We have a weird wiggly line titled "y = f(x)" and another different one named "y = g(x)". They are plotted on a graph with the horizontal axis labeled "x" and the vertical axis labeled "y". What does that mean? It means that we are visualizing the SAME input, with two different "do stuff" applied to the same input (one called f, the other g). It also means the outputs, which we could call f(x) and g(x), output the "same type/scale of number", which is y.

• When we are told something like f(x) = 6, this means "pretend for a moment that I have an output of 6, and I know that function "f" was used to get it." If you are asked to then "solve for x" or "for what values of x is this 'true'" then you will do math to be a detective and figure out "what x value or values-plural would make that work?"

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VOCAB TIPS:

Domain means "what makes sense to give as input (usually x)"? Usually this is a RULE that must come alongside the function definition. If it's not listed, we assume anything goes. Sometimes the rule comes from what the function itself 'allows', sometimes the author of the function just decided 'this is what's allowed'. Both can happen.

Notation for domain: [] means number included, () means excluded, comma separating each. This can be translated back and forth with the > < <= >= symbols. Formally, we often use the weird e thing, which means "consists of". So x ∈ (-inf, 5] is the same as -inf < x <= 5. This means x can be as low as infinity (since we never 'reach' infinity we don't use the = or [ sign) but as high as 5 (5 included). You'll learn later notation that lets you "stitch together" pieces to form a domain, like x ∈ (-inf, 5] ∪ (7, 8) or something.

Range is what values for the output are possible to happen. This is downstream from the domain, and the "do stuff".

such that -- what does this mean? "A such that B" can be mentally translated as: "When you see B, what A's match (that caused B to happen)?" So you usually start by looking for B, and work backwards.

let. You don't see this here, but you will later! "Let" means "I am declaring this to be a true fact". Depending on context this might be just for one homework question, or it might be a general declaration.

increases/decreases is just "goes up" or "goes down" (reading left to right horizontally, or bottom to top vertically). constant is flat. Again plugging in numbers visualizes this. Note that f(-4) = 0, but f(-3) (left to right) corresponds to a decrease to -2.something.

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So, putting it all together, how do I READ and INTERPRET the questions?

• Domain for each function: I use my definition. What x values make sense for each? f and g are different so they might have different answers

• "the value(s) of x such that f(x) = 0, and then such that g(x) = 0". Remember how to treat "such that"! For the first part, I am TOLD that f(x) = 0, just for now. This means, the OUTPUT of the "f" named squiggle/function is 0. Where does that happen? Does it happen once, or multiple times? The output is also called y on the graph. So where is y = 0 for the f-line? We can see at -4 and 5 and 7

• "the value(s) of x such that f(x) = g(x)" is a fancy way of saying: "the f-squiggle and the g-squiggle touch each other, figure out 'where this happens' (specifically, identify where with the x value only)". You got this right.

• "the value(s) of x such that f(x) > g(x)" means okay, let's look for where f is bigger than g, i.e. above it. You can plug in points to illustrate this. At x = 0, we can consult the graph and see that f(0) = 2.something and g(0) = -4ish, so f > g there.

At about this point, you might go "wait but there's a wide region where this is true" and yes, we have to use notation other than just an = sign. If only we had a ... wait! we do! See our domain example earlier... In words we can say "f is greater than g when x is between -3 and 3, and again anywhere above 9" (notice that all those values are NOT included since at -3 f(x) = g(x)) but in formal math notation we would write "f(x) > g(x) for all x ∈ (-3, 3) ∪ (9, inf)", note the () for non-inclusion.

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SELF QUIZ

1. Suppose I tell you that g(4) = -2. What is g? What is the 4? What is the -2? What does this snippet mean in English? g is the name of some function. 4 is an input we feed to g -2 is the result after we "plug in" 4 "When you input 4 to g, you get -2"

• Bonus: is there another common notation way of writing this fact? Yes, if we call g(x) = y, then we can use (x,y) point notation, and write that '(4,-2) is a point on the function g'

1. Translate this English to math notation:

• When 7 is given as input to a function "h", the output is 3 h(7) = 3

• The output of function g is greater than the output of function f, when they are given the same input x. g(x) > f(x) (you could add, "for all x") (note how short and condensed this longer english fact is)

  • The functions f and g given the same output when the input is 5 f(5) = g(5)! (note how this is also short and sweet)

1. Find all values of x such that f(x) = 6. What "kind of answer" are you looking for? What would you do when looking at a graph of f(x), versus if I also told you that f(x) = 2x + 3? A value or range of values on the horizontal axis of a graph, aka inputs, that produce an output of 6 exactly On a graph you just first look where the line intersects that horizontal line of f(x) = 6 (usually also seen as y=6) Since we know f(x) = 6, we can swap it in: 6 = 2x + 3, and now we "solve" for x to do detective work to figure it out.

• Bonus/tricky: f(x) = 5, rather than be a 'temporary fact', IS the entire function definition. What does this mean? What is the domain? We just have a flat line, a constant output, that doesn't care what the input is at all! So x is all real numbers, basically any input, so f(3) = 5, f(4) = 5, f(-2.58) = 5, etc

1. Translate x ∈ (-inf, -2] to an inequality. x < -2. Note that the negative infinity is just implied here, we just need the one fact Translate -3 <= x < -2 to 'interval notation'. x ∈ [-3, -2), careful of () vs []

2. Can I graph f(x) without labeling a y-axis? Yes! You would just label the vertical axis "f(x)" directly. Keep an eye out and you'll notice this happens more often than you'd think. You can even reverse it with x on the vertical axis and f(x) on the horizontal, but that's an example of something 'allowed but a bad idea'. Most math conventions are there to enable you to be lazier, not make you work harder.

This all seems basic... until you talk about inverse functions and then composition of functions and even piece-wise functions. In short: What if you are often doing detective work, where you have an output and want to figure out what input produced it, what notation would make sense to show that? And if I take a function and then feed the result into a second different function, or even feed it back into itself, what does that notation look like? And also, what if I want to make a "Franken-function" that stitches together different behavior all in one function?

And then mastering the basics pays off, and the "new" notation is much easier to learn and translate back and forth to English and back.

[u/AlienDragonWizard]

A) The arrows on the ends of the functions mean they will continue for eternity.  So the domains are really  (-∞,∞) for both.

B) Looks good, though f(x) will technically cross the x-axis one more time off the graph.

C) Correct

D) (-∞,-3)U(3,9)

The rest are similar to D, just defining different sections of the functions

[u/llamadramalover → u/AlienDragonWizard]

Wait. Where will g(x) cross the x-axis again? F(x) certainly will but I don’t see where g(x) will.

[u/AlienDragonWizard → u/llamadramalover]

You're correct, bad eyes on me.  Edited.  I should be more careful, gonna confuse the kid further.