Educational

Check Bijective Function Check One-One and Onto 🎥 Video Explanation 📝 Question Let \( f:\mathbb{N} \to \mathbb{Z} \) be defined by: \[ f(n)= \begin{cases} \dfrac{n-1}{2}, & \text{if } n \text{ is odd} \\ -\dfrac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] A. neither one-one nor onto B. one-one but not onto C. onto […]

Range of Permutation Function Find Range of Function 🎥 Video Explanation 📝 Question \[ f(x) = {}^{7-x}P_{x-3} \] A. {1, 2, 3, 4, 5} B. {1, 2, 3, 4, 5, 6} C. {1, 2, 3, 4} D. {1, 2, 3} ✅ Solution 🔹 Step 1: Validity Conditions For permutation \( ^nP_r \): \[ n \ge

Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Given \( f:[0,\infty) \to \mathbb{R} \), \[ f(x)=\frac{x}{x+1} \] A. one-one and onto B. one-one but not onto C. onto but not one-one D. neither one-one nor onto ✅ Solution 🔹 Step 1: Check Injective Let \(x_1 \ne x_2\). \[ f(x)=\frac{x}{x+1} \] is

Determinant Function Type Check One-One / Onto 🎥 Video Explanation 📝 Question Let \(M\) be the set of all \(2 \times 2\) matrices with real entries. Function: \[ f : M \to \mathbb{R}, \quad f(A)=|A| \] A. one-one and onto B. neither one-one nor onto C. one-one not one-one D. onto but not one-one ✅

Greatest Integer Function Type Check One-One / Onto 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \) be defined as \[ f(x)=[x]^2+[x+1]-3 \] where \([x]\) is the greatest integer ≤ \(x\). (a) many-one and onto (b) many-one and into (c) one-one and into (d) one-one and onto ✅ Solution 🔹 Step 1: Use

Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( A = B = \{x \in \mathbb{R} : -1 \le x \le 1\} \). Function: \[ f(x) = x|x| \] A. injective but not surjective B. surjective but not injective C. bijective D. none of these ✅ Solution 🔹 Step 1:

Bijective Quadratic Function Check Bijective Condition 🎥 Video Explanation 📝 Question Given \( f(x) = -x^2 + 6x – 8 \), find the correct domain \(A\) and codomain \(B\) such that \(f\) is bijective. A. \(A = (-\infty, 5]\), \(B = (-\infty, 1]\) B. \(A = [-3, \infty)\), \(B = (-\infty, 1]\) C. \(A =

Check Bijective Function Check One-One and Onto 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} – \{-b\} \to \mathbb{R} – \{1\} \) be defined by \[ f(x)=\frac{x+a}{x+b}, \quad a \ne b \] Choose the correct option: A. one-one but not onto B. onto but not one-one C. both one-one and onto D. none of these

Check Function Type Check One-One / Onto 🎥 Video Explanation 📝 Question Given \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=2^x + 2^{|x|} \] Find the correct option: A. one-one and onto B. many-one and onto C. one-one and into D. many-one and into ✅ Solution 🔹 Step 1: Case-wise Expression Case 1: \(x \ge 0\)

Find f(x) from Equation Find Function f(x) 🎥 Video Explanation 📝 Question If \( f : A \to B \) is defined by \[ 3^{f(x)} + 2^{-x} = 4 \] and \(f\) is bijective, find \(f(x)\). ✅ Solution 🔹 Step 1: Isolate \(f(x)\) \[ 3^{f(x)} = 4 – 2^{-x} \] — 🔹 Step 2: Take