Educational

Check Function Type Check One-One / Onto 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R}\setminus\{n\} \to \mathbb{R} \) be defined by \[ f(x)=\frac{x-m}{x-n}, \quad m \ne n \] A. one-one onto B. one-one into C. many-one onto D. many-one into ✅ Solution 🔹 Step 1: Check Injective Assume \(f(x_1)=f(x_2)\): \[ \frac{x_1-m}{x_1-n}=\frac{x_2-m}{x_2-n} \] Cross multiply: \[ […]

Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \) be defined by \[ f(x)=\frac{e^{|x|}-e^{-x}}{e^x+e^{-x}} \] A. bijection B. injection only C. surjection only D. neither ✅ Solution 🔹 Step 1: Case-wise Simplification Case 1: \(x \ge 0\) \[ |x|=x \] \[ f(x)=\frac{e^x – e^{-x}}{e^x + e^{-x}}

Check Bijective Function Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\left[-\tfrac12,\tfrac12\right] \to \left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right] \) be defined by \[ f(x)=\sin^{-1}(3x-4x^3) \] A. bijection B. injection but not a surjection C. surjection but not an injection D. neither ✅ Solution 🔹 Step 1: Key Identity \[ 3x – 4x^3 = \sin(3\theta) \quad \text{if

Check Function Type Check One-One / Onto 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \) be defined by \[ f(x)=(x-1)(x-2)(x-3) \] A. one-one but not onto B. onto but not one-one C. both one-one and onto D. neither one-one nor onto ✅ Solution 🔹 Step 1: Nature of Function \(f(x)\) is a

Bijective Quadratic Function Find Set B for Bijectivity 🎥 Video Explanation 📝 Question Let \( f:[2,\infty) \to B \) be defined by \[ f(x)=x^2-4x+5 \] Find \(B\) such that \(f\) is bijective. A. \(\mathbb{R}\) B. \([1,\infty)\) C. \([4,\infty)\) D. \([5,\infty)\) ✅ Solution 🔹 Step 1: Convert to Vertex Form \[ f(x)=x^2-4x+5 \] \[ = (x-2)^2

Range of Function Find Set A for Surjection 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to A \) be defined by \[ f(x)=\frac{x^2}{x^2+1} \] Find \(A\) such that \(f\) is surjective. A. \(\mathbb{R}\) B. \([0,1]\) C. \((0,1]\) D. \([0,1)\) ✅ Solution 🔹 Step 1: Analyze Function \[ f(x)=\frac{x^2}{x^2+1} \] Rewrite: \[ f(x)=1-\frac{1}{x^2+1} \] —

Function x|x| Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \(A = \{x : -1 \le x \le 1\}\). \[ f(x)=x|x| \] A. a bijection B. injective but not surjective C. surjective but not injective D. neither injective nor surjective ✅ Solution 🔹 Step 1: Case-wise Form For \(x \ge 0\): \[

Check Bijective Functions Check Which Functions are Bijective 🎥 Video Explanation 📝 Question Let \(A = \{x : -1 \le x \le 1\}\). Which of the following functions \(A \to A\) are bijections? A. \(f(x)=\frac{x}{2}\) B. \(g(x)=\sin\left(\frac{\pi x}{2}\right)\) C. \(h(x)=|x|\) D. \(k(x)=x^2\) ✅ Solution 🔹 Option A: \(f(x)=\frac{x}{2}\) Range: \[ [-1/2, 1/2] \] Not equal

Check Bijective Functions Check Which Functions are Bijective 🎥 Video Explanation 📝 Question Which of the following functions \(f:\mathbb{Z} \to \mathbb{Z}\) are bijections? A. \(f(x)=x^3\) B. \(f(x)=x+2\) C. \(f(x)=2x+1\) D. \(f(x)=x^2+x\) ✅ Solution 🔹 Option A: \(f(x)=x^3\) Injective ✔️ (strictly increasing) Not onto ❌ (e.g., 2 is not a cube of any integer) ⇒ Not

Injective Function Logic Problem Find \(f^{-1}(1)\) 🎥 Video Explanation 📝 Question Let \(f\) be an injective map from \(\{x,y,z\}\) to \(\{1,2,3\}\). Exactly one of the following is true: \(f(x)=1\) \(f(y)\ne 1\) \(f(z)\ne 2\) Find \(f^{-1}(1)\). ✅ Solution 🔹 Step 1: Injective ⇒ Bijective Since domain and codomain have same number of elements, \(f\) is bijective.