Converting Functions from x to y

[ozone]

I just need help turning a function of x into a function of y.

I'll give you an example and maybe you can explain it and expand upon the idea.

Thanks in advance.

y = x^2 -5x (see a very simple function! I just can't figure out how to convert it).

or y = x^2 - 2x

 

[Mark44]

I just need help turning a function of x into a function of y.

I'll give you an example and maybe you can explain it and expand upon the idea.

Thanks in advance.

y = x^2 -5x (see a very simple function! I just can't figure out how to convert it).

or y = x^2 - 2x

What you're asking is not clear, but I think you are talking about finding the inverse function. In your first example, y = f(x) = x² - 5x. Here y is a function of x. It is not, however, a one-to-one function, so the inverse will not be a function.

To solve for x in terms of y, complete the square.
y = x² - 5x
==> y = x² - 5x + (5/2)² - (5/2)² = (x - 5/2)² - (5/2)²

Can you take it from there?

To make the inverse a function, restrict the domain of the original function to make it one-to-one.

 

[dynamicsolo]

Beware, however: your function y = x² - 5x (the graph of which is a parabola) assigns two different values of x to the same value of y . So when you "invert" this to get x = "something" , you will get a plus-or-minus sign, which tells us that we get back two functions of y , each corresponding to half of the original parabola.

This is always a problem with even powers of x in a function; reversing the direction to return from y to x gives two possible values of x which could have led to that value of y , and a "rule" which assigns one value of y to two values of x is NOT a function. That's why we need to split your result into two parts. (We don't have this problem with odd powers of x .)

 

[ozone]

Thank you for your answers.
Yes I was looking to complete the square, and now that I've seen this it is quite a bit more complicated then I expected.