1. ## Rules on Functions

I've been looking around on the internet, but I can't make much sense of the questions I have with the definitions for function that I'm getting, and I have answers but I'm not sure how correct they are.

I'm looking at the graph of a function.

1. What is the maximum number of times that the graph can intersect the y-axis.

I'm thinking that a function could only cross the y-axis once, because for the input 0 a function can only have 1 output.

2. Can the graph intersect the x-axis an infinite number of times?

My question is can a function be of an infinite degree? Does this scenario make sense?

3. If the function is a polynomial, how can you predict the number of times it might cross the x-axis?

The answer that I believe is right for this is that the polynomial function will have as many roots as the number of its degree, providing that all the roots are real numbers.

I feel like I know the first and third answers, but am unsure about the second and I think I could use a bit of conceptual help. Thanks to anyone who can help explain these scenarios to me!

2. Originally Posted by mikedwd
I've been looking around on the internet, but I can't make much sense of the questions I have with the definitions for function that I'm getting, and I have answers but I'm not sure how correct they are.

I'm looking at the graph of a function.

1. What is the maximum number of times that the graph can intersect the y-axis.

I'm thinking that a function could only cross the y-axis once, because for the input 0 a function can only have 1 output.
you are correct

2. Can the graph intersect the x-axis an infinite number of times?

My question is can a function be of an infinite degree? Does this scenario make sense?
consider the sine function or the cosine function or the tangent function or the cotangent function or ...

3. If the function is a polynomial, how can you predict the number of times it might cross the x-axis?

The answer that I believe is right for this is that the polynomial function will have as many roots as the number of its degree, providing that all the roots are real numbers.
the roots include multiplicity. so it can have as many roots as its degree, or it can have less (even if all roots are real), or it can have no real roots at all.

3. Alright, thanks a lot for your help!

I completely forgot about trigonometric functions and multiplicity! Thanks for catching me

greatly appreciated, -Mike