# Functions Problem 1

Before you proceed further, check your main outline. A proof that a function is
bijective normally divides into two parts, one proving onto and one proving one-to-one.
Only in very special situations does it work well to
deal with both properties simultaneously.

### One-to-one

Now, check the subparts. For one-to-one, you should have something like:

Let x and y be real numbers in \([\frac{1}{4},3]\).
Suppose that f(x) = f(y).

[Use the definition of f to turn f(x)=f(y) into algebra.]

......

So x=y.

### Onto

For onto, you should have something like:

Let y be an element in the interval [0,1].
Consider the value x = [some expression in terms of y].

Then f(x) = ....

So y = f(x).

How do you figure out which value to "consider" for x? Well, you
need to work backwards from the goal on your scratch paper. Set
up the equation y=f(x). Then apply the definition of f to turn
f(x) into algebra. Now solve for x.

For this proof, typical of many onto proofs, the work on your scratch
paper will be backwards. When you copy it into the above
outline, you'll need to reverse the order.

### Moving forwards

When you have a proof working with this outline, you are ready for
further testing.