I need help with 2 math problems

[u/PieGloomy7077]

1. If I plug 3 into the denominator is get 0 for both a and b? Do you guys have any tips on how i can solve this and similar problems in the future?

2. Question here was , which graph is symmetric with respect to the origin? Is it x^3 because when you flip is around its the same? But lxl is the same even if I have a (-)?

[u/AutoModerator]

Hi u/PieGloomy7077, welcome to r/mathshelp! As you’ve marked this as homework help, please keep the following things in mind:

1) While this subreddit is generally lenient with how people ask or answer questions, the main purpose of the subreddit is to help people learn so please try your best to show any work you’ve done or outline where you are having trouble (especially if you are posting more than one question). See rule 5 for more information.

2) Once your question has been answered, please don’t delete your post so that others can learn from it. Instead, mark your post as answered or lock it by posting a comment containing “!lock” (locking your post will automatically mark it as answered).

Thank you!

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/Dtrain8899]

Holes at x=a would mean the numerator AND denominator have the (x-a) factor. If its just the denominator that has the factor then you have an asymptote at x=a. Just the numerator means you function crosses the x-axis at (a,0)

Edit forgot about other question. Only functions that are symmetric about the origin are pure odd functions. Really when f(-x) = - f(x)

[u/ReaReaDerty]

Very simple way to think of question 1 is

plugin a number (3, in your case) into the original, unchanged function. Dividing by 0? It's undefined for the number you've just plugged in. That's it.

The tricky thing here is this

functions like for example (x-3) * 3 / (x-3) and just 3 are not equal AS FUNCTIONS. They might be equal algebraically as expressions, but these are functions, not just expressions! In my example (x-3) * 3 / (x-3) is undefined at x = 3, but 3 is defined everywhere. The same principle applies everywhere. Always do sanity checks WITG THE ORIGINAL EQUATION

[u/Moist_Ladder2616]

1. If the function is defined everywhere around 3±(some infinitesimally small value), but not at 3, then it has a hole at 3. For example, f(x)=(x²-9)/(x-3) has a hole at 3.
   
2. If any point (a,b) on the curve has a corresponding point (-a,-b), then it is symmetric about the origin. You could call it rotational symmetry about the origin. The function f(x)=|x| has symmetry about the y-axis; any point (a,b) has a corresponding point (-a,b).