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. […]

Find Range and Pre-Images of f(x) = x² Find Range and Pre-Images of \(f(x)=x^2\) Question: A function \(f : \mathbb{R} \to \mathbb{R}\) is defined by $$ f(x)=x^2 $$ Determine: (i) Range of \(f\) (ii) \(\{x : f(x)=4\}\) (iii) \(\{y : f(y)=-1\}\) Solution Given: $$ f(x)=x^2 $$ (i) Range of \(f\) Since the square of every

Piecewise Defined Function on Real Numbers Piecewise Defined Function on Real Numbers Question: If a function \(f : \mathbb{R} \to \mathbb{R}\) is defined by $$ f(x)= \begin{cases} 3x-2, & x0 \end{cases} $$ Solution The given function is a piecewise defined function. Different algebraic expressions are used for different values of \(x\). The function is defined

Find Range and Pre-Image of a Function Find Range and Pre-Image of a Function Question: Let \( A = \{-2,-1,0,1,2\} \) and \( f : A \to \mathbb{Z} \) be a function defined by \( f(x) = x^2 – 2x – 3 \). Find: (i) Range of \(f\), i.e. \(f(A)\) (ii) Pre-image of \(6\), \(-3\)

Difference Between Relation and Function Difference Between Relation and Function Question: What is the fundamental difference between a relation and a function? Is every relation a function? Solution A relation is a set of ordered pairs that shows the relationship between two sets. A function is a special type of relation in which every input

Define a Function as a Correspondence Between Two Sets Define a Function as a Correspondence Between Two Sets Question: Define a function as a correspondence between two sets. Solution A function is a rule or correspondence between two sets that assigns each element of one set to exactly one element of another set. If $A$

Define a Function as a Set of Ordered Pairs Define a Function as a Set of Ordered Pairs Question: Define a function as a set of ordered pairs. Solution A function is a special type of relation represented as a set of ordered pairs. A function from a set $A$ to a set $B$ is

Write R in Roster Form if x − y is Odd Write R in Roster Form if x − y is Odd Question Let \[ A=\{1,2,3,5\}, \quad B=\{4,6,9\} \] and \( R \) be a relation from \( A \) to \( B \) defined by \[ R=\{(x,y): x-y \text{ is odd}\} \] Write \(

Find Sets A and B if (x,1), (y,2), (z,1) are in A × B Find Sets A and B if (x,1), (y,2), (z,1) are in A × B Question Let \( A \) and \( B \) be two sets such that \[ n(A)=3 \quad \text{and} \quad n(B)=2. \] If \[ (x,1),(y,2),(z,1)\in A\times B, \]

Find the Domain and Range of R = {(x, y) : 2x + y = 8} Find the Domain and Range of R = {(x, y) : 2x + y = 8} Question If \[ R=\{(x,y): x,y\in W,\ 2x+y=8\}, \] then write the domain and range of \( R \). Solution Given, \[ 2x+y=8 \]