Which Relation is Not a Function? Which Relation is Not a Function? Question: Let $$ A=\{p,q,r,s\} $$ and $$ B=\{1,2,3\} $$ Which of the following relations from \(A\) to \(B\) is not a function? (i) $$ R_1=\{(p,1),(q,2),(r,1),(s,2)\} $$ (ii) $$ R_2=\{(p,1),(q,1),(r,1),(s,1)\} $$ (iii) $$ R_3=\{(p,1),(q,2),(p,2),(s,3)\} $$ (iv) $$ R_4=\{(p,2),(q,3),(r,2),(s,2)\} $$ Solution A relation is a […]

Find Pre-Images of a Function Find Pre-Images of a Function Question: If $$ f:\mathbb{R}\to\mathbb{R} $$ is defined by $$ f(x)=x^2+1 $$ then find $$ f^{-1}\{17\} $$ and $$ f^{-1}\{-3\} $$ Solution Given: $$ f(x)=x^2+1 $$ Find \(f^{-1}\{17\}\) $$ x^2+1=17 $$ $$ x^2=16 $$ $$ x=\pm4 $$ Therefore, $$ f^{-1}\{17\}=\{-4,4\} $$ Find \(f^{-1}\{-3\}\) $$ x^2+1=-3 $$

Find the Range of a Function Using Highest Prime Factor Find the Range of a Function Using Highest Prime Factor Question: Let $$ A=\{12,13,14,15,16,17\} $$ and $$ f:A\to \mathbb{Z} $$ be a function given by $$ f(x)=\text{highest prime factor of }x $$ Find the range of \(f\). Solution Find the highest prime factor of each

Determine Functions from X to Y and Find Their Range Determine Functions from X to Y and Find Their Range Question: Let $$ X=\{1,2,3,4\} $$ and $$ Y=\{1,5,9,11,15,16\} $$ Determine which of the following sets are functions from \(X\) to \(Y\) and find the range of each function. (i) $$ f_1=\{(1,1),(2,11),(3,1),(4,15)\} $$ (ii) $$ f_2=\{(1,1),(2,7),(3,5)\}

Find the Range of Functions Find the Range of Functions Question: If \(f\), \(g\), \(h\) are three functions defined from \(\mathbb{R}\) to \(\mathbb{R}\) as follows: $$ f(x)=x^2 $$ $$ g(x)=\sin x $$ $$ h(x)=x^2+1 $$ Find the range of each function. Solution For $$ f(x)=x^2 $$ the square of every real number is always non-negative.

Are Two Functions Equal? Are Two Functions Equal? Question: Let $$ f:\mathbb{R}\to\mathbb{R} $$ and $$ g:\mathbb{C}\to\mathbb{C} $$ be two functions defined as $$ f(x)=x^2 $$ and $$ g(x)=x^2 $$ Are they equal functions? Solution Two functions are equal if they have: same domain same codomain same rule of correspondence Here, $$ f:\mathbb{R}\to\mathbb{R} $$ and $$

Write Relation as Ordered Pairs and Check Function Write Relation as Ordered Pairs and Check Whether it is a Function Question: $$ \{(x,y): x+y=3,\ x,y\in\{0,1,2,3\}\} $$ Solution Given: $$ x+y=3 $$ where $$ x,y\in\{0,1,2,3\} $$ Taking values satisfying \(x+y=3\): $$ 0+3=3 $$ $$ 1+2=3 $$ $$ 2+1=3 $$ $$ 3+0=3 $$ Therefore, the ordered pairs

Write Relation as Ordered Pairs and Check Function Write Relation as Ordered Pairs and Check Whether it is a Function Question: $$ \{(x,y): y>x+1,\ x=1,2 \text{ and } y=2,4,6\} $$ Solution For \(x=1\), $$ y>1+1 $$ $$ y>2 $$ So, \(y=4,6\) Ordered pairs are: $$ (1,4),\ (1,6) $$ For \(x=2\), $$ y>2+1 $$ $$ y>3

“`html Write Relation as Ordered Pairs and Check Function Write Relation as Ordered Pairs and Check Whether it is a Function Question: Write the following relation as sets of ordered pairs and determine whether it is a function: $$ \{(x,y): y=3x,\ x\in\{1,2,3\},\ y\in\{3,6,9,12\}\} $$ Solution Given relation: $$ y=3x $$ where $$ x\in\{1,2,3\} $$ and

Properties of Log Function f(x) = logₑx Properties of Log Function \(f(x)=\log_e x\) Question: Let \(f : \mathbb{R}^+ \to \mathbb{R}\), where \(\mathbb{R}^+\) is the set of all positive real numbers, be such that $$ f(x)=\log_e x $$ Determine: (i) the image set of the domain of \(f\) (ii) \(\{x : f(x)=-2\}\) (iii) whether \(f(xy)=f(x)+f(y)\) holds.