Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 136

Functions. Is g(x) etc, the OUTPUT?

In a function is the part to the left hand side of the equals sign the OUTPUT?

For example, regards g(x) = x+2, is the following correct;

g(x) = OUTPUT

x = INPUT

x+2 = RELATIONSHIP

I know that if, regards the above, x=4, then we put 4 in and get 6 out, so 4 is the input and 6 is the output; but is g(x) also considered to be the OUTPUT? Or, in this case, g(4)?

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Functions. Is g(x) etc, the OUTPUT?

g(x)=y
g(4)="y, when x=4"
output=result-of-relationship-formula, given input

Bob

Administrator

Registered: 2010-06-20

Posts: 10,168

Re: Functions. Is g(x) etc, the OUTPUT?

Yes, that looks good to me. I think of a function as a box with one or more inputs and a single output.

Happy New Year!

Bob

---

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

paulb203

Member

Registered: 2023-02-24

Posts: 136

Re: Functions. Is g(x) etc, the OUTPUT?

Thanks, guys.
One definition of function I came across online was;
"A function is a rule that maps a number to another unique number."

Q. Do you both agree with that defnition?
Q. Regards the 'unique number' part; I just came across a function question the answer to which was two numbers, either or, (a quadratic equation was involved; how does that fit with the 'unique number' part?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,168

Re: Functions. Is g(x) etc, the OUTPUT?

That definition is ok but it is also possible to have several inputs.

See https://www.khanacademy.org/math/multiv … 0function.

All the definitions I have seen insist that the function is 'well defined' which means you have to know what the output is so that doesn't allow multi variable outputs.

see https://www.mathsisfun.com/sets/function.html

I'd like to see that function question. 

I wouldn't expect a question such as "Evaluate the following function when x = 3" to have more than one answer.

But that Khan page goes on to talk about functions that have two vector components. I don't think that contradicts the one output rule though, as the vector example they give leads to a single vector with two components, not two vectors.

Bob

---

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

paulb203

Member

Registered: 2023-02-24

Posts: 136

Re: Functions. Is g(x) etc, the OUTPUT?

Thanks, Bob.

I'll check out those links when I have time.

In the meantime, just a quick question.

Regards our saying that y=f(x), when looking for the inverse function; would it be accurate to say, in general (not just with functions), that, for example, y=x, or y=x+2, or whatever, that this is actually shorthand for y(x)=x, or y(x)=x+2, or whatever?

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 164

Re: Functions. Is g(x) etc, the OUTPUT?

I wonder how we can see this case:

An event occurs whenever Time = arcsine (0.5) is satisfied.

Perhaps we need to differentiate between a function and its inverse.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,168

Re: Functions. Is g(x) etc, the OUTPUT?

would it be accurate to say, in general (not just with functions), that, for example, y=x, or y=x+2, or whatever, that this is actually shorthand for y(x)=x, or y(x)=x+2

Yes, that's ok.

An event occurs whenever Time = arcsine (0.5) is satisfied.

Perhaps we need to differentiate between a function and its inverse.

To be a function there must only be one output. 

If each output can only occur once for every input (such as y = x + 2) the function is described as 1:1. ie 1 input 1 output.

But a function may also be many:1 such as y = x^2.  Here x=5 gives y = 25, and also x = -5 gives y = 25.

It's a function because we can compute a single output for every input.

SINE is another example of a many:1 function. For every angle we can find SINE(angle) but many angles can give the same SINE. eg. SINE(30) = SINE(150) = SINE(390) and so on.

If you try to construct an inverse for a many to 1 function you run into a problem ... which output from the original function should you choose as the input for the inverse?

You can get around this problem by restricting the domain of the inverse so that only one input exists.

For  arcsine the usual restriction is that the angle must be between -90 and + 90. That way arcsine has a unique value for each input.

There a bit about it here:
https://www.mathsisfun.com/sets/function-inverse.html

And more about mappings here:

https://www.mathsisfun.com/sets/injecti … ctive.html

Bob

---

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

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 164

Re: Functions. Is g(x) etc, the OUTPUT?

You can get around this problem by restricting the domain of the inverse so that only one input exists.

Thank you.
In other words, in the full definition of a function, the domain in which it is valid is also specified.

paulb203

Member

Registered: 2023-02-24

Posts: 136

Re: Functions. Is g(x) etc, the OUTPUT?

Thanks, guys.