# Function's Fun f(x^2−2016x)=f(x)⋅x+2016

• MHB
• cooltu
In summary, the domain of the function is all real numbers except for 2016, as it would result in division by zero. The function is not one-to-one and can result in the same output for different inputs. The range of the function is all real numbers and it is undefined at x = 2016. The function can be rewritten as f(x) = x^2-2016x.

#### cooltu

f(x^2*2016x) = f(x)x+2016
Then f(2017) = ?

Isn't "x^2*2016x" the same as "2016x^3"?

Thanks, skeeter. That sounds more reasonable.

f(x^2- 2016x)= f(2017) if x^2- 2016x= 2017. That is the same as x^2- 2016x- 2017= (x+ 1)(x- 2017)= 0 so either x= -1 or x= 2017.

So either f(2017)= -1(f(-1))+ 2016 or f(2017)= 2017 f(2017)+ 2016.

For the latter, -2016= 2016 f(2017) so f(2017)= -1.

## 1. What is the domain of the function?

The domain of the function is all real numbers except for 2016, since the expression x^2-2016x would result in division by zero.

## 2. Is the function one-to-one?

No, the function is not one-to-one because for two different inputs, x and y, the function can result in the same output f(x) = f(y) if x^2-2016x = y^2-2016y.

## 3. What is the range of the function?

The range of the function is all real numbers, as the function can take on any value depending on the input x.

## 4. Is there a specific value for which the function is undefined?

Yes, the function is undefined at x = 2016, as mentioned in the domain.

## 5. Can this function be simplified or rewritten?

Yes, this function can be rewritten as f(x) = x^2-2016x, since f(x) is equal to both sides of the original equation.