Can somebody please explain what’s going here I have no clue and didn’t see any videos on this. Thanks so much
What’s f(-1)? Let its value be x. What is f(x)?
f(something) implies we’re asking f(value of x - coordinate) = what value of y .
In the ETS math review there is a full section on it! (Algegbra - 2.9 Graphs of Functions)
Thanks, so I could follow the explanation but what I didn’t get is why first we look up the x value and then we use y value. Let me take a look at the math review, maybe I read it but didn’t “record” it in my mind. Thanks
thanks
wow the math review is really concise, it’s not really explaining nested functions…
Sorry not sure if I follow, can you read my logic and see if it makes sense?
So we’ll focus on f(-1) first where -1 represents the x value. So where x is -1, what is the output? Output is expressed in terms of y. So where x = -1, we find a y of 2. Thus, f(-1)=2
Question: this is always the case? So your input of x leads to a y output?
So then we can rewrite the question as
f(f(-1)) = ?
f(2) = ?
in other words: where x = 2, what is the y value
f(2) = 1 and thus D
Nested function means function inside a function and the rule to solve it to start from the inside and move outside.
\overbrace{f(\underbrace{f(-1)}_{\text{Inside}})}^{\text{outside}}
Thus, first you need to find the value of y at the f(-1) means x coordinate = -1. Now, can you look at the graph and see what that value might be?
yep,exactly !
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That is ETS for you, they will not mention each question type that you may encounter explicitly but what the question is really testing is you’re ability to understand what is mean by f(something) and if you read about functions and coordinate plane from Math Review then that’s a given.
can you also give some words to these kind of problems? i understand the theory kind of (not all of it) but it’s just that because i cannot TALK about it to someone, I don’t grasp it.
i know the theory behind odd functions and even functions but i never knew that this kind of question means a reflection over x and y value and has a relation. So odd functions is when all exponents are odd and even functions when exponents are even. i found a video but it’s explained where unclear, do you have suggestions?
thanks so much for your help i appreciate it
Have you seen the video “Even and Odd functions” on PrepSwift? That could help you here.
I did but it’s not going as deep for me to thoroughly understand what’s happening.
if I raise a negative odd/even number to an odd power then what will be the result ?
if I raise a negative odd/even number to an even power then what will be the result ?
Same question with positive
if I raise a positive odd/even number to an even power then what will be the result ?
if I raise a positive odd/even number to an even power then what will be the result ?
for eg: Option A → f(x)= \dfrac{x^3}{(x^2+1)} Now, the denominator part : (x^2+1) will always be positive because x^{\text{even}} = \text{ positive}
Now, the numerator part x^3 will be positive for positive value of x but will be negative for negative values that is : 5^3 = 125 (\text{ positive}) \text{ but} -5^3 = -125(\text{ negative})
The question ask : f(-x) = -f (x)
so, for negative value of x ; our option A numerator will be negative and the denominator will be positive because it’s raise to even power so positive in all cases.
Thus, option A become \dfrac{-\text{ numerator}}{\text{ denomintor}}= - \dfrac{\text{ numerator}}{\text{ denomintor}} = - \left(\dfrac{x^3}{(x^2+1)}\right). Now , this the left side of the question .i.e f(-x)
Now, the right side is -f(x). And what it means is : - \left( \dfrac{x^3}{(x^2+1)} \right).
As right hand and left hand are the same we can choose A!
You can do the same analysis for other options! As you get more familiar with such problems you wouldn’t even have to write much of the stuff and can eyeball it !
These are the same concepts that are in ETS math review / PrepSwift ! It’s just that the way it’s formulated makes it harder to identify! Once your fundamentals are clear then you’ll be able to see what a question is really trying to ask!
Thank you so much for takin the time to explain this to me. I have my test end of November and sometimes it seems like I’m going backward instead of forward. Discussing on the forum helps a lot so thanks again. I’ll need to reread your thoughts a few times for it to be engrained.
So new question. When it says f(-x) = f(x) Greg says in his video “a negative input means nothing happens to the output” which I understand because of the example that is given. BUT if we take it literally, am I correct that f(-x) = f(x) says “a negative input will result in the same output as when you input the same input but then positive”. Trying to rephrase in my own words to understand it properly.
So then the odd function confuses me, because the generalized version says: f(-x) = -f(x)
So inputting x, is the same as inputting x but then the minus before the f means it becomes negative. But can you explain me in words what it means… What does it mean to put a negative sign BEFORE the function. Is it somewhere along the line of
inputting -x is the same as inputting x BUT multiplied by -1
Indeed.
The result is multiplied by -1.
In an even function, when you input -x instead of x, you get the same result. In an odd function, you get the negative of the result instead.
I’d recommend you play around with f(x) = x^2 and g(x) = x^3. Identify which of the two would be odd/even/neither, and justify.
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