Math Is Fun Forum

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Reciprocal Function Properties

Let f(x) = 1/x be the reciprocal function.

1. Why is the domain and range of this function the set of all nonzero real numbers?

2. Why is this function odd?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,198

Re: Reciprocal Function Properties

Arhhh! Here is the function y = 1/x. Looks like you got this and the previous question muddled.

Using graph sketching techniques:

When x tends to - infinity y tends to zero from negative values.

When x tends to infinity y tend to zero from positive values.

As x approaches zero y tends to infinity when x>0 and - infinity when x<0

So the graph has two elements: from the left starts close to zero and gets more and more negative as x approaches zero from the left.

It is undefined at x = 0. **

After that it 'comes down' from infinity and gradually tends to zero as x gets bigger.

x and y are never zero so are excluded from the domain and range.

(x, 1/x) lies on the graph and so does (-x, -1/x) which means f(-x) = - f(x).

Bob

** My advanced level teacher said the two sections of the graph meet 'round the back of infinity'  !!! This is, of course, complete rubbish; even if you have a smile on your face as you say it.  She wasn't really up to the needed standard for an A level teacher.  She made other errors, one of which helped me to get a place at university! They asked about mathematical induction and I gave her definition. The interviewers looked puzzled and asked if that was what I thought too. I answered "no" and gave the correct definition and explained why I was right and she was wrong.  I learnt much later that my answer was one of the reasons they offered me a place!

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 549

Re: Reciprocal Function Properties

Arhhh! Here is the function y = 1/x. Looks like you got this and the previous question muddled.

Using graph sketching techniques:

When x tends to - infinity y tends to zero from negative values.

When x tends to infinity y tend to zero from positive values.

As x approaches zero y tends to infinity when x>0 and - infinity when x<0

So the graph has two elements: from the left starts close to zero and gets more and more negative as x approaches zero from the left.

It is undefined at x = 0. **

After that it 'comes down' from infinity and gradually tends to zero as x gets bigger.

x and y are never zero so are excluded from the domain and range.

(x, 1/x) lies on the graph and so does (-x, -1/x) which means f(-x) = - f(x).

Bob

** My advanced level teacher said the two sections of the graph meet 'round the back of infinity'  !!! This is, of course, complete rubbish; even if you have a smile on your face as you say it.  She wasn't really up to the needed standard for an A level teacher.  She made other errors, one of which helped me to get a place at university! They asked about mathematical induction and I gave her definition. The interviewers looked puzzled and asked if that was what I thought too. I answered "no" and gave the correct definition and explained why I was right and she was wrong.  I learnt much later that my answer was one of the reasons they offered me a place!

Thank you for sharing your story about how you landed the position at the university through someone else's ignorance.