Show that if $g(f(x)) $ is one-to-one and $f$ is onto, then $g$ is one-to-one
46863
Show that $g$ is injective
45416
linear transformation $T$ is onto if and only if $T^*$ is one to one
121729
Prove without using graphing calculators that $f: \mathbb R\to \mathbb R,\,f(x)=x+\sin x$ is both one-to-one, onto (bijective) function.
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
58642
If $g \circ f$ is the identity function, then which of $f$ and $g$ is onto and which is one-to-one?
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
55175
Is the function $f(x) = x^2 - 2x$ one-to-one?
61782
How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
70034
Showing that this set of functions is a group.
40609
Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent
43956
Differentiating $f(x)^{g(x)}$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
79281
Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general
70912
If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
46748
If $g \circ f$ is surjective, show that $f$ does not have to be surjective?
48765
Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.