Express Product As A Function

[harpazo]

The sum of two numbers is 40. Express their product as a function of one of the numbers.

Solution:

Let x and y represent the two numbers.

x + y = 40

Their product means to multiply x•y.

We want y expressed in terms of x.

Let y = f(x).

The book goes on to say that we should solve x + y = 40 for y.

x + y = 40

y = 40 - x

I now plug into xy.

f(x) = x(40 - x)

f(x) = 40x - x^2

Question:

Would it make a difference if I decided to solve x + y = 40 for x?

Let me see.

x = 40 - y

Then xy becomes (40 - y)y.

I could say f(y) = (40 - y)y or
40y - y^2.

Is this true? Can this change by made?

What about the graph of the two functions? Do they look exactly the same on the xy-plane?

In other words, does f(x) = f(y) on the xy-plane?

 

[Dr.Peterson]

The graphs would not be in the xy-plane. What you have is a function P = f(x), in the "xP-plane" if you will, and another function P = f(y), in the "yP-plane".

They are the same function, just expressed in terms of different variables: f(x) = 40x - x^2 , and f(y) = 40y - y^2. Remember that the variable used is irrelevant; the function is all about what it does to whatever input you give it, in this case to multiply the input by 40 and subtract the square of the input. For this reason, it is appropriate to use the same name, f.

Now, looking back at the problem, we have two numbers, x and y, which play symmetrical roles -- you could swap x and y and everything about them would be unchanged. It makes no difference which one you call x and which you call y. That's why you get the same function of either variable. In a different problem, that might not happen -- for instance, if it were x - y = 40 rather than x + y = 40. In this case, they would be distinguished as "the larger number" and "the smaller number", and the product would be a different function of each.

 

[harpazo]

The graphs would not be in the xy-plane. What you have is a function P = f(x), in the "xP-plane" if you will, and another function P = f(y), in the "yP-plane".

They are the same function, just expressed in terms of different variables: f(x) = 40x - x^2 , and f(y) = 40y - y^2. Remember that the variable used is irrelevant; the function is all about what it does to whatever input you give it, in this case to multiply the input by 40 and subtract the square of the input. For this reason, it is appropriate to use the same name, f.

Now, looking back at the problem, we have two numbers, x and y, which play symmetrical roles -- you could swap x and y and everything about them would be unchanged. It makes no difference which one you call x and which you call y. That's why you get the same function of either variable. In a different problem, that might not happen -- for instance, if it were x - y = 40 rather than x + y = 40. In this case, they would be distinguished as "the larger number" and "the smaller number", and the product would be a different function of each.

Informative reply.