Injectivity, surjectivity and image set

Text

Answer each request:

• (a) draw a Venn diagram (sets and arrows) of an injective but not surjective function;
• (b) draw the graph of a function of your choice that is injective but not surjective, giving domain, codomain and image;
• (c) draw a Venn diagram of a surjective but not injective function;
• (d) draw the graph of a function of your choice that is surjective but not injective, giving domain, codomain and image;
• (e) find the image set of: (i) $y=-x^2+x+1$; (ii) the ellipse with centre $(2;-3)$ and semi-axes $a=2,\ b=3$; (iii) the hyperbola $-x^2+2y^2=1$.

Solution

(a) E.g. $A=\{1,2\}\to B=\{p,q,r\}$ with $1\mapsto p$, $2\mapsto q$: distinct images (injective), but $r$ is not reached (not surjective).

(b) E.g. $f(x)=e^x$: injective (increasing), image $(0;+\infty)\subsetneq\mathbb{R}$, hence not surjective. Domain $\mathbb{R}$, codomain $\mathbb{R}$, image $(0;+\infty)$.

(c) E.g. $A=\{1,2,3\}\to B=\{p,q\}$ with $1\mapsto p$, $2\mapsto q$, $3\mapsto q$: every element of $B$ reached (surjective), but $2,3$ share an image (not injective).

(d) E.g. $f(x)=x^3-x$: image $\mathbb{R}$ (surjective) but not injective. Domain $\mathbb{R}$, codomain $\mathbb{R}$, image $\mathbb{R}$.

(e) (i) parabola with maximum at $x=\tfrac12$, $y_{max}=\tfrac54$: image $\left(-\infty;\tfrac54\right]$. (ii) the ordinate varies by $\pm b=\pm3$ about $-3$: image $[-6;0]$. (iii) $-x^2+2y^2=1\Rightarrow y^2\ge\tfrac12$: image $\left(-\infty;-\tfrac{\sqrt2}{2}\right]\cup\left[\tfrac{\sqrt2}{2};+\infty\right)$.

$\boxed{\text{(i) }\left(-\infty,\tfrac54\right];\ \text{(ii) } [-6,0];\ \text{(iii) } |y|\ge\tfrac{\sqrt2}{2}}$