Thread: One-one function

Find the derivative of where and ,

Hence show that is a one-one function

I have found the derivative of h(x) to be but am confused on how i could use this to show is one-one

If a function is either always increasing or always decreasing then it is one to one,

oh... so u mean that the gradient is always decreasing or increasing, then it is one to one?

Originally Posted by HallsofIvy

If a function is either always increasing or always decreasing then it is one to one,

sorry but i didnt quite get what you meant

Do you understand what "one to one" means? A function, f, is one to one if and only if f(x)= f(y) implies x= y. If f is increasing, we can use "proof by contradiction": Suppose f(x)= f(y) with . Then either x> y so that f(x)> f(y) or x< y f(x)< f(y), either contradicting f(x)= f(y). The same argument works if f is decreasing.

Originally Posted by Punch

oh... so u mean that the gradient is always decreasing or increasing, then it is one to one?

No, I mean that if the function is increasing or decreasing (it gradient (derivative) is non-negative or non-positive) then it is one to one.

Originally Posted by HallsofIvy

No, I mean that if the function is increasing or decreasing (it gradient (derivative) is non-negative or non-positive) then it is one to one.

Actually, to be technically precise, the derivative would have to be positive or negative; it could only be zero at isolated points and not on any interval.

For a function to be one-to-one, the function must pass the vertical line test and the horizontal line test for all values of x.