How to know if equation is one-to-one

[intenzxboi]

i know for a equation to be 1-1 they have to pass the horizontal line test.

so say for example
(-8 + x) ^4

i know this is not a one to one because it gives me a parabola.

The problem that I am having is being able to explain in a equation.

MY SOLUTION

First derive the equation so that i know where it is increasing/decreasing

4(-8+x)^3

so i take -8+x>0 is increasing
x>8 <----- I am sure that's correctnow to find it decreasing you flip the sign

and i take -8+x<0 is decreasing?

but then i also get x<8

What I'm i doing wrong??So if I am doing this right then that means that for a function to be one to one.. It either has to be increasing OR decreasing?? but cannot be both?

 

[Mark44]

First off, it makes no sense to say that an equation is one-to-one. A function can be one-to-one.

Second, an equation has an = sign in it, and makes a statement about two expressions being equal.

Your first example, (x - 8)^4 is not an equation. What you probably mean is y = (x - 8)^4, which is an equation, and is the equation of a function. This can also be represented as f(x) = (x - 8)^4. The graph of this function is not a parabola, but it has a similar U shape.

Let's take the derivative: f'(x) = 4(x - 8)^3

The function f is increasing when f'(x) > 0, or when (x - 8)^3 > 0, which is when x > 8.

The function f is decreasing when f'(x) < 0, or when (x - 8)^3 < 0, which is when x - 8 < 0. IOW, when x < 8.

So the function is decreasing on the interval ($-\infty$, 8) and is increasing on the interval (8, $\infty$). This means that for some y values, there are two x values, so this function is not one-to-one.

You could also show this by picking an appropriate y-value and showing that there are two x values.

For example, if (x - 8)^4 = 1, then x - 8 = 1 or x - 8 = -1, so x = 9 or x = 7.

 

[epenguin]

You are perhaps confusing yourself by using unnecessary machinery that you will need for other cases that are not so obvious.

f is one-to-one if f(a) = f(b) implies a = b.

From what you say about the function f(x) = (x - 8)⁴ you seem to know well enough its shape. I think I glimpsed somewhere there is something called the horizontal line test. If necessary draw pictures, or calculate some numbers. I think it is pretty obvious where this function has its minimum. If you can't see it, calculate f(a) for some number a, then think whether any other number, which would be your b, gives you the same output number.

(A quibble that does not affect this issue is that this function is not a parabola - that term is reserved for a second degree function like (x - 8)². If you plot it you will notice a difference to parabolas you have seen. However for the essential properties you are concerned with here it is working just like a parabola.)