Let $X = \mathbb{R}$ and $Y = \left \{ x \in \mathbb{R}\mid x ≥ 1 \right \}$. Define $G : X → Y$ by $G(x) = e^{x^2}$. Prove that $G$ is onto.
67234
The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$
61375
Meaning of $\mathbb{R}[x]$
85612
Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
70912
If $g\circ f$ is injective and $f$ is surjective then $g$ is injective
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
119156
$(a,b)=d \overset{?}{\implies} (a^3,b^3)=d^3$
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
138297
Let $G = GL_2\mathbb (R)$. Show that $T$ is a subgroup of $G$
78698
Prove if $f$ has a right inverse function: $f\circ g=id_x$ $\iff$ $f$ is onto $Y$
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
103801
$xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
118348
Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$
11798
If $f \circ g$ is onto then $f$ is onto and if $f \circ g$ is one-to-one then $g$ is one-to-one
72251
$g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
27188
Bijection between $\mathbb{R}$ and $\mathbb{R}/\mathbb{Q}$