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