Recent questions in Relations and Functions

Find if the given operation has identity: $\;\; a \ast b = a^2 + b^2$

asked Nov 21, 2012 by vaishali.a

2 answers

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows: $\;\; a \ast b = a-b$ . Find which of the binary operations are commutative and which are associative.

asked Nov 21, 2012 by vaishali.a

1 answer

Let $\ast$ be the binary operation on $N$ defined by $a \ast b=H.C.F.$ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on $N$ ?

asked Nov 21, 2012 by vaishali.a

1 answer

Is $\ast$ defined on the set $\{1,2,3,4,5\}$ by $a\ast b=L.C.M$ .of $a$ and $b$ a binary operation?

asked Nov 21, 2012 by vaishali.a

1 answer

Let $\ast$ be the binary operation on $N$ given by $a\ast b=L.C.M\,.of\,a\,and\,b$ . Find $\begin{array}{1 1}(i)\;\; 5 \ast7,\; 20 \ast 16 & (ii)\;\; Is\; \ast \;commutative?\\(iii)\;\; Is\; \ast \;associative? & (iv)\;\; Find\, the\,identity \, of \ast \, in N\\(v)\;\; Which \, elements\, of \, N\, are\, invertible\, for\, the\, operaation\,\ast ? & \;\end{array}$

asked Nov 21, 2012 by vaishali.a

2 answers

Let $\ast '$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a \ast ' b = H.C.F$ of a and b. Is the operation same as the operation $\ast$ defined in the table below? $\begin{matrix} *&1&2&3&4&5 \\ 1&1&1&1&1&1 \\ 2&1&2&1&2&1 \\ 3&1&1&3&1&1 \\ 4&1&2&1&4&1 \\ 5&1&1&1&1&5 \end{matrix}$

asked Nov 21, 2012 by vaishali.a

1 answer

Consider the binary operation $\wedge$ on the set $\{1, 2, 3, 4, 5\}$ defined by $a \wedge b = min \{a, b\}$ . Write the multiplication table of the operation $\wedge$ .

asked Nov 20, 2012 by vaishali.a

1 answer

For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,ab\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,ab\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,ab\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, ab= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,ab\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, ab= \frac {a} {b+1} & \;\end{array}$

asked Nov 20, 2012 by vaishali.a

5 answers

Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this - On $\; Z^+,\,$ defined $\,$ by $\; ab= a-b$

asked Nov 19, 2012 by vaishali.a

1 answer

Let $f : R - \{ - \frac {4} {3} \} \to R$ be a function defined as $f(x)= \frac {4x} {3x+4}$ .The inverse of $f$ is the map $g$ : Range $f \to R - \{ - \frac {4} {3} \}$ given by

asked Nov 19, 2012 by vaishali.a

1 answer

If $f: R\to R$ be given by $f(x)=(3-x^3)^\frac {1}{3}$ , then $fof(x)$ is

asked Nov 19, 2012 by vaishali.a

1 answer

Let $f : X \to Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$ , i.e., $(f^{-1})^{-1} = f$ .

asked Nov 16, 2012 by vaishali.a

1 answer

Consider $f:\{1,2,3\} \to\{a,b,c\}$ given by $f(1)=a, \,f(2)=b$ and $f(3)=c$ . Find $f^{-1}$ and show that $(f^{-1})^{-1} = f$ .

asked Nov 15, 2012 by vaishali.a

1 answer

Let $f : X \to Y$ be an invertible function. Show that $f$ has unique inverse.

asked Nov 15, 2012 by vaishali.a

1 answer

Consider $f:R_+ \to [\;\text{–5}, \infty )$ given by $f(x)=9x^2 +6x$ - $5$ .Show that $f$ is invertible with $f^{-1} (y) = \bigg(\frac{(\sqrt{y+6}) -1} { 3}\bigg)$

asked Nov 15, 2012 by vaishali.a

1 answer

Consider $f:R_+ \to [4,\infty)$ given by $f(x)=x^2 +4$ . Show that $f$ is invertible with the inverse $f^{-1}$ of $f$ given by $f^{-1}(y) = \sqrt {y-4}$ ,where $R_+$ is the set of all non-negative real numbers.

asked Nov 14, 2012 by vaishali.a

1 answer

Consider $f:R \to R$ given by $f(x)=4x+3$ . Show that $f$ is invertible. Find the inverse of $f$

asked Nov 14, 2012 by vaishali.a

1 answer

Show that $f : [-1,1]$ $\rightarrow R$ , given by $f(x) =\frac {x } { (x+2)}$ is one-one. Find the inverse of of the $f : [-1,1] \rightarrow$ Range $f$ .

asked Nov 14, 2012 by vaishali.a

1 answer

State with reason whether following functions have inverse: (iii) $h: \{2,3,4,5,\} \to \{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13) \}$

asked Nov 9, 2012 by vaishali.a

1 answer

If $f(x) = \frac { (4x+3) } { (6x-4) }, x \neq \frac {2} {3}$ , show that $f(x) =x$ , for all $x \neq \frac {2} {3}$ . What is the inverse of $f$

asked Nov 9, 2012 by vaishali.a

1 answer

Find $gof$ and $fog$ , if (i) $f(x) = |\;x\;| \, and \, g(x) = |\;5x-2\;|$

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f,\, g\, and\, h$ be functions from $R\, to\, R.$ Show that $(f+g) oh = foh + goh$

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f:\{1,3,4\} \to \{1,2,5\}$ and $g:\{1,2,5\}\to\{1,3\}$ be given by $f = \{(1, 2), (3, 5), (4, 1)\}$ and $g = \{(1, 3), (2, 3), (5, 1)\}$ . Write down $gof$ .

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f : R \to R$ be defined as $f (x) = 3x$ . Choose the correct answer:

asked Nov 9, 2012 by vaishali.a

1 answer

Let $f: R \to R$ , be defined as $f(x) = x^4$ . Choose the correct answer.

asked Nov 9, 2012 by vaishali.a

1 answer