'Showing' that a function is one-to-one

[MarkSA]

Hello,

I have a number of problems in my homework that have the first step as 'Show that the function is one-to-one'. I know that I can determine if a function is one-to-one by doing the horizontal line test, and also by looking for functions where all exponents are odd, but i'm not sure how to actually 'show' it or what they are asking me to do. Is there a procedure (probably fairly simple) to 'show' this?

 

[pka]

A function f is one-to-one if and only if f(a)= f(b) implies that a=b.
In set theory we say that a function is one-to-one if and only if no two pairs have the same second term (that happens to be the vertical line test).

 

[MarkSA]

A function f is one-to-one if and only if f(a)= f(b) implies that a=b.

Do you know a way that I could 'show' this as the homework asks though? Just plugging in a or b for into the functions and setting them equal will always come out to a=b, won't it? even if I have f(x) = x^2, which is not one-to-one, if I do f(a) = f(b), I get a^2 = b^2, +-a = +-b

For instance, one question has the function:
f(x) = sqrt(x^3 + x^2 + x + 1)

Another has the function: f(x) = sqrt(x - 2)

 

[pka]

Here the last one.
\(\displaystyle \begin{array}{l} f(x) = \sqrt {x - 2} \\ & f(a) = f(b) \\ & \sqrt {a - 2} = \sqrt {b - 2} \\ & a - 2 = b - 2 \\ & a = b \\ \end{array}\)