Converting Functions: A Guide from x to y

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In summary: I just need help turning a function of x into a function of y.In summary, you need to find the inverse of a function that is not a one-to-one, and you need to do this by restricting the domain of the original function.
  • #1
ozone
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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
 
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  • #2
ozone said:
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.
 
  • #3
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 .)
 
  • #4
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.
 

1. What is a function?

A function is a mathematical relationship between two or more variables, where an input (or inputs) is transformed into an output (or outputs) according to a specific rule or set of rules.

2. What is the purpose of using functions?

Functions are used to organize and streamline complex calculations or processes into smaller, more manageable parts. They also allow for the reuse of code, making programming more efficient and less prone to errors.

3. How do you define a function?

A function is defined by providing a name, a set of input parameters, and a set of instructions or statements to be executed. The function can then be called or invoked by passing in values for the input parameters.

4. What is the difference between a function declaration and a function expression?

A function declaration is a statement that declares a function and its name, while a function expression is an assignment of a function to a variable. Function declarations are hoisted to the top of the scope, while function expressions are not.

5. Can a function return multiple values?

Yes, a function can return multiple values by using an array or an object to store and return the values. Alternatively, a function can use the arguments object to access all input parameters and return them in an array or object.

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