Question

Prove that the given function is surjective.

(a)$\mathit{f}\mathbf{:}\mathit{R}\mathbf{\to }\mathit{R}\mathbf{;}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}$

(b)$\mathit{f}\mathbf{:}\mathit{Z}\mathbf{\to }\mathit{Z}\mathbf{;}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{4}$

(c)$\mathbf{}\mathit{f}\mathbf{:}\mathit{R}\mathbf{\to }\mathit{R}\mathbf{;}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{5}$

(d)$\mathit{f}\mathbf{:}\mathit{Z}\mathbf{\times }\mathit{Z}\mathbf{\to }\mathit{Q}\mathbf{;}\mathit{f}\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}\mathbf{=}\mathit{a}\mathbf{/}\mathit{b}\mathbf{\hspace{0.33em}}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathbf{\hspace{0.33em}}\mathit{b}\mathbf{\ne }\mathbf{0}\mathbf{\hspace{0.33em}}\mathit{a}\mathit{n}\mathit{d}\mathbf{\hspace{0.33em}}\mathbf{0}\mathbf{\hspace{0.33em}}\mathit{w}\mathit{h}\mathit{e}\mathit{n}\mathbf{}\mathit{b}\mathbf{=}\mathbf{0}$

Short answer

Thus,(a)$f\left(x\right)=x^3$ is not surjective.

(b)$f\left(x\right)=x-4$ is surjective.

(c)$f\left(x\right)=-3x+5$ is surjective.

(d)$f(a,b)=a/b$ is surjective.

Step-by-step solution

(a) Showing that f(x)=x3 is surjective

It is given that $f\left(x\right)=x^3$. Let $y=x^3$, then

role="math" localid="1659172635405" $x={}^3\sqrt{y}$

Here, $x,y\in R.$.

If take negative value of$y=-2$ then it gives,

$$\begin{gathered} x={}^3\sqrt{y} \\ ={}^3\sqrt{\left(-2\right)} \end{gathered}$$

Which does not give real number.

Hence,role="math" localid="1659172683432" $f\left(x\right)=x^3$ is not surjective

(b) Showing that  f(x)=x-4 is surjective

It is given that $f\left(x\right)=x-4$. Let $y=x-4$, then

$$\begin{gathered} y=x+4 \\ x=y+4 \end{gathered}$$

Here, $x,y\in Z.$. Then, put y=1 then it gives,

$$\begin{gathered} x=1+4 \\ =5 \end{gathered}$$

Thus, the value is also $x\in Z.$

Hence, $f\left(x\right)=x-4$is surjective.

(c) Showing that f(x)=-3x+5  is surjective

It is given that $f\left(x\right)=-3x+5$. Let $y=-3x+5$then,

$x=\frac{y-5}{-3}$

Here, $x,y\in Z$. Then, put$y=1$ then it gives,

$$\begin{gathered} x=\frac{1-5}{-3} \\ =\frac{4}{3} \end{gathered}$$

Thus, the value is also $x\in Z.$.

Hence,$f\left(x\right)=-3x+5$ is surjective

(d) Showing that f(a,b)=ab is surjective

It is given that $f(a,b)=\frac{a}{b}$. Let $y=\frac{a}{b}$.

When $b\ne 0$

$a=by$

Then, it is surjective.

When $b=0$

$a=0$

Then, it is also surjective.

Hence,role="math" localid="1659173254022" $f(a,b)=\frac{a}{b}$ is surjective