Relations and Functions

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

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

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

$f:\left[-1,1\right]\to R$ is given as $f\left(x\right)=\frac{x}{x+2}$

$\begin{array}{l}Let,f\left(x\right)=f\left(y\right).\\ \Rightarrow \frac{x}{x+2}=\frac{y}{y+2}\\ \Rightarrow xy+2x=xy+2y\\ \Rightarrow 2x=2y\\ \Rightarrow x=y\end{array}$

$\therefore$ f is a one-one function.

It is clear that $f:\left[-1,1\right]\to$ Range f is onto.

$\therefore$ $f:\left[-1,1\right]\to$ Range f is one-one onto and therefore, the inverse of the function:

$f:\left[-1,1\right]\to$ Range f exists.

Let g: Range $f\to \left[-1,1\right]$ be the inverse of f.

Let y be an arbitrary element of range f.

Since $f:\left[-1,1\right]\to$ Range f is onto, we have:

$\begin{array}{l}\Rightarrow y=\frac{x}{x+2}\\ \Rightarrow xy+2y=x\\ \Rightarrow x\left(1-y\right)=2y\\ \Rightarrow x=\frac{2y}{1-y},y\ne 1\\ g\left(y\right)=\frac{2y}{1-y},y\ne 1\\ Now,\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(\frac{x}{x+2}\right)=\frac{2\left(\frac{x}{x+2}\right)}{1-\frac{x}{x+2}}=\frac{2x}{x+2-x}=2\frac{x}{2}=x\\ \left(fog\right)\left(y\right)=f\left(g\left(y\right)\right)=f\left(\frac{2y}{1-y}\right)=\frac{\frac{2y}{\left(1-y\right)}}{\left(\frac{2y}{1-y}\right)+2}=\frac{2y}{2y+2-2y}=\frac{2y}{2}=y\\ \therefore gof=I_{-1,1},and,fog-I_{Range,f}\\ \therefore f^{-1}=g\\ \therefore f^{-1}\left(y\right)=\frac{2y}{1-y},y\ne 1\end{array}$

It is given that $f\left(x\right)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$

$\begin{array}{l}\left(fof\right)\left(x\right)=f\left(f\left(x\right)\right)=f\left(\frac{4x+3}{6x-4}\right)\\ =\frac{4\left(\frac{4x+3}{6x-4}\right)+3}{6\left(\frac{4x+3}{6x-4}\right)-4}=\frac{16x+12+18x-12}{24x+18-24x+16}=\frac{34x}{34}=x\end{array}$

Therefore $fof\left(x\right)=x$ for all $x\ne \frac{2}{3}$

$\Rightarrow fof=1$

Hence, the given function f is invertible and the inverse of f is itself.

To prove:

$\begin{array}{l}\left(f+g\right)oh=foh+goh\\ consider:\\ \left(\left(f+g\right)oh\right)\left(x\right)\\ =\left(f+g\right)\left(h\left(x\right)\right)\\ =f\left(h\left(x\right)\right)+g\left(h\left(x\right)\right)\\ =\left(foh\right)\left(x\right)+\left(goh\right)\left(x\right)\\ =\left\{\left(foh\right)+\left(goh\right)\right\}\left(x\right)\\ \therefore \left(\left(f+g\right)oh\right)\left(x\right)=\left\{\left(foh\right)+\left(goh\right)\right\}\left(x\right),\forall x\in R\\ Hence,\left(f+g\right)oh=foh+goh\end{array}$

To prove

$\begin{array}{l}\left(f.g\right)oh=\left(foh\right).\left(goh\right)\\ Consider\\ \left(\left(f.g\right)oh\right)\left(x\right)\\ =\left(f.g\right)\left(h\left(x\right)\right)\\ =f\left(h\left(x\right)\right).g\left(h\left(x\right)\right)\\ =\left(foh\right)\left(x\right).\left(goh\right)\left(x\right)\\ =\left\{\left(foh\right).\left(goh\right)\right\}\left(x\right)\\ \therefore \left(\left(f.g\right)oh\right)\left(x\right)=\left\{\left(foh\right).\left(goh\right)\right\}\left(x\right),\forall x\in R\\ Hence,\left(f.g\right)oh=\left(foh\right).\left(goh\right)\end{array}$

The functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} are defined as

f = { (1, 2), (3, 5), (4, 1)} and g = { (1, 3), (2, 3), (5, 1)}.

gof (1) = g (f (1) = g (2) = 3 [f (1) = 2 and g (2) = 3]

gof (3) = g (f (3) = g (5) = 1 [f (3) = 5 and g (5) = 1]

gof (4) = g (f (4) = g (1) = 3 [f (4) = 1 and g (1) = 3]

$\therefore$ gof = { (1, 3), (3, 1), (4, 3)}