Relations and Functions

The binary operation? on the set {1, 2, 3, 4, 5} is defined as a? b = min {a, b} &mn For E; a, b? {1, 2, 3, 4, 5}.

Thus, the operation table for the given operation? can be given as:

 (i) On Z, * is defined by a * b = a − b.

It can be observed that 1 * 2 = 1 − 2 = 1 and 2 * 1 = 2 − 1 = 1.

∴1 * 2 ≠ 2 * 1; where 1, 2 ∈ Z

Hence, the operation * is not commutative.

Also we have:

(1 * 2) * 3 = (1 − 2) * 3 = −1 * 3 = −1 − 3 = −4

1 * (2 * 3) = 1 * (2 −

(i) On Z⁺, * is defined by a * b = a − b.

It is not a binary operation as the image of (1, 2) under * is 1 * 2 = 1 − 2= −1 ∉ Z⁺.

(ii) On Z⁺, * is defined by a * b = ab.

It is seen that for each a, b ∈ Z⁺, there is a unique element ab in Z⁺.

This means that * carries each pair (a, b) to a unique eleme

It is given that $f:R-\left\{-\frac{4}{3}\right\}\to R$ is defined as $f\left(x\right)=\frac{4x}{3x-4}$

Let y be an arbitrary element of Range f.

Then, there exists $x\in R-\left\{-\frac{4}{3}\right\}$ such that $y=f\left(x\right)$

$\Rightarrow y=\frac{4x}{3x+4}$

$\Rightarrow 3xy+4y=4x$ let us define g: Range $f\to R-\left\{-\frac{4}{3}\right\}as,g\left(y\right)=\frac{4y}{4-3y}$

$\begin{array}{l}\Rightarrow x\left(4-3y\right)=4y\\ Now,\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(\frac{4x}{3x+4}\right)\\ \Rightarrow x=\frac{4y}{4-3y}\\ =\frac{4\left(4\frac{x}{3x+4}\right)}{4-3\left(\frac{4x}{3x+4}\right)}=\frac{16x}{12x+16-12x}=\frac{16x}{16}=x\\ And,\left(fog\right)\left(y\right)=f\left(g\left(y\right)\right)=f\left(\frac{4y}{4-3y}\right)\\ =\frac{4\left(\frac{4y}{4-3y}\right)}{3\left(\frac{4y}{4-3y}\right)+4}=\frac{16y}{12y+16-12y}=\frac{16y}{16}=y\\ \therefore gof=I_{R-\left(-\frac{4}{3}\right)}and,fog=I\_"Range,f"\end{array}$

Thus, g is the inverse of f i.e., $f^{-1}=g.$

Hence, the inverse of f is the map g: Range $f\to R-\left\{-\frac{4}{3}\right\}$ which is given by

$g\left(y\right)=\frac{4y}{4-3y}$

The correct answer is

$f:R\to R$ is given as $f\left(x\right)={\left(3-x^{\wedge }3\right)}^{\wedge }\left(1/3\right)$

$\begin{array}{l}f\left(x\right)={\left(3-x^3\right)}^{\frac{1}{3}}\\ \therefore fof\left(x\right)=f\left(f\left(x\right)\right)=f\left({\left(3-x^3\right)}^{\frac{1}{3}}\right)={\left[3-{\left({\left(3-x^3\right)}^{\frac{1}{3}}\right)}^3\right]}^{\frac{1}{3}}\\ ={\left[3-\left(3-x^3\right)\right]}^{\frac{1}{3}}={\left(x^3\right)}^{\frac{1}{3}}=x\\ \therefore fof\left(x\right)=x\end{array}$

The correct answer is C.

Let f: X → Y be an invertible function. 

Then, there exists a function g: Y → X such that gof = IX and fog = IY. 

Here, f⁻¹ = g. 

Now, gof = IX and fog = IY

⇒ f⁻¹ of = IX and fof⁻¹ = IY

Hence, f⁻¹ : Y → X is invertible and f is the inverse of f⁻¹ 

i.e., (f−1)⁻¹  = f.

$f\left(1\right)=a,f\left(2\right)b,and,f\left(3\right)=c$

If we define $g:\left\{a,b,c\right\}\to \left\{1,2,3\right\}as,g\left(a\right)=1,g\left(b\right)=2,g\left(c\right)=3,$ then we have:

  $\begin{array}{l}\left(fog\right)\left(a\right)=f\left(g\left(a\right)\right)=f\left(1\right)=a\\ \left(fog\right)\left(b\right)=f\left(g\left(b\right)\right)=f\left(2\right)=b\\ \left(fog\right)\left(c\right)=f\left(g\left(c\right)\right)=f\left(3\right)=c\\ And\\ \left(gof\right)\left(1\right)=g\left(f\left(1\right)\right)=f\left(a\right)=1\\ \left(gof\right)\left(2\right)=g\left(f\left(2\right)\right)=f\left(b\right)=2\\ \left(gof\right)\left(3\right)=g\left(f\left(3\right)\right)=f\left(c\right)=3\\ \therefore gof=I_{X}and,fog=I_Y\\ Where,X=\left\{1,2,3\right\},and,Y=\left\{a,b,c\right\}.\end{array}$

Thus, the inverse of f exists and $f^{-1}=g.$

$\therefore f^{-1}:\left\{a,b,c\right\}\to \left\{1,2,3\right\}$ is given by,

$\begin{array}{l}{f}^{-1}\left(a\right)=1,f^{-1}\left(b\right)=2,f^{-1}\left(c\right)=3\\ \end{array}$

Let us now find the inverse of $f^{-1}$ i.e., find the inverse of g.

If we define $h:\left\{1,2,3\right\}\to \left\{a,b,c\right\}as$

$h\left(1\right)=a,h\left(2\right)=b,h\left(3\right)=c$ , then we have

$\begin{array}{l}\left(goh\right)\left(1\right)=g\left(h\left(1\right)\right)=g\left(a\right)=1\\ \left(goh\right)\left(2\right)=g\left(h\left(2\right)\right)=g\left(b\right)=2\\ \left(goh\right)\left(3\right)=g\left(h\left(3\right)\right)=g\left(c\right)=3\\ And\\ \left(hog\right)\left(a\right)=h\left(g\left(a\right)\right)=h\left(1\right)=a\\ \left(hog\right)\left(b\right)=h\left(g\left(b\right)\right)=h\left(2\right)=b\\ \left(hog\right)\left(c\right)=h\left(g\left(c\right)\right)=h\left(3\right)=c\\ \therefore goh=I_{X}and,hog=I_Y\\ Where,X=\left\{1,2,3\right\},and,Y=\left\{a,b,c\right\}.\end{array}$

Thus, the inverse of g exists and $g^{-1}=h\Rightarrow {\left(f^{-1}\right)}^{-1}=h.$

It can be noted that h=f.

Hence, ${\left(f^{-1}\right)}^{-1}=f.$

Let $f:X\to Y$ be an invertible function.

Also, suppose f has two inverses (say g₁ and g₂ ).

Then, for all y ∈ Y, we have:

$\begin{array}{l}fo{g}_1\left(y\right)=I_y\left(y\right)=fo{g}_2\left(y\right)\\ \Rightarrow f\left(g_1\left(y\right)\right)=f\left(g_2\left(y\right)\right)\end{array}$  [f is invertible => f is one-one]

$\Rightarrow g_1=g_2$  [g is one-one]

Hence, f has a unique inverse.

$f:R_{+}\to [4,\infty )$ is given as $f\left(x\right)=x^2+4$ .

One-one:

$\begin{array}{l}Let,f\left(x\right)=f\left(y\right).\\ \Rightarrow x^2+4=y^2+4\\ \Rightarrow x^2=y^2\\ \Rightarrow x=y\left[as,x=y\in R\right]\end{array}$

$\therefore f$ is a one-one function.

$f:R\to R$ is given by,

$\begin{array}{l}f\left(x\right)=4x+3\\ One-one:\\ Let,f\left(x\right)=f\left(y\right).\\ \Rightarrow 4x+3=4y+3\\ \Rightarrow 4x=4y\\ \Rightarrow x=y\end{array}$

$\therefore$ f is a one-one function.

Onto:

$\begin{array}{l}For,y\in R,let,y=4x+3.\\ \Rightarrow x=\frac{y-3}{4}\in R\end{array}$

Therefore, for any $y\in R$ , there exists $x=\frac{y-3}{4}\in R$ such that

$f\left(x\right)=f\left(\frac{y-3}{4}\right)=4\left(\frac{y-3}{4}\right)+3=y$

$\therefore$ f is onto.

Thus, f is one-one and onto and therefore, $f^{-1}$ exists.

Let us define $g:R\to R$ by $g\left(x\right)=\frac{y-3}{4}$

$\begin{array}{l}Now,\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(4x+3\right)=\frac{\left(4x+3\right)-3}{4}=x\\ \left(fog\right)\left(y\right)=f\left(g\left(y\right)\right)=f\left(\frac{y-3}{4}\right)=4\left(\frac{y-3}{4}\right)+3=y-3+3=y\\ \therefore gof=fog=I_R\end{array}$

Hence, f is invertible and the inverse of f is given by

$f^{-1}=g\left(y\right)=\frac{y-3}{4}$