Now, let's check a few detailed issues.
For the onto proof, notice that the input value x has to be chosen from the domain of f. This isn't all of \(\mathbb{R}\) but only the interval \([\frac{1}{4},3]\). Somewhere in your proof, you'll need to argue that this is true. You know that your output value y comes from the interval [0,1]. So, on your scratch paper, figure out why this implies that your choice for x is in \([\frac{1}{4},3]\). Incorporate this explanation into your proof.
Step back and look at your proof. Did you put in enough connector words, or does it look like a bare sequence of equations? Are the words written out in English or have you used obscure patterns of dots and arrows?
Yes, you are the only person who will read this draft. But if you don't practice writing proofs out properly, you'll have trouble writing them quickly under test conditions. This goes double for folks whose first language isn't English.
When you have a draft proof that makes it through the above tests, check it against the annotated solutions.