Educational

Check Function \(f(x)=\dfrac{2x+3}{x-3}\) on Rational Numbers 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Q}\setminus\{3\}\to\mathbb{Q},\quad f(x)=\frac{2x+3}{x-3} \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ \frac{2x_1+3}{x_1-3}=\frac{2x_2+3}{x_2-3} \] Cross multiply: \[ (2x_1+3)(x_2-3)=(2x_2+3)(x_1-3) \] Expand: \[ 2x_1x_2-6x_1+3x_2-9=2x_1x_2-6x_2+3x_1-9 \] Simplify: \[ -6x_1+3x_2=-6x_2+3x_1 \] \[ […]

Check Function \(f(x)=\sin^2x+\cos^2x\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=\sin^2x+\cos^2x \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Simplify Function Using identity: \[ \sin^2x+\cos^2x=1 \] So: \[ f(x)=1 \quad \text{for all } x\in\mathbb{R} \] 🔹 Step 2: Check Injection (One-One) A function

Check Function \(f(x)=x^3-x\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=x^3-x \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) A function is one-one if different inputs give different outputs. Take: \[ x=-1,\quad x=0,\quad x=1 \] Then: \[ f(-1)=(-1)^3-(-1)=0 \] \[

Check Function \(f(x)=x^3+1\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=x^3+1 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3+1=x_2^3+1 \] So: \[ x_1^3=x_2^3 \] Cube function is strictly increasing on real numbers. Therefore:

Check Function \(f(x)=\sin x\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=\sin x \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) A function is one-one if different inputs give different outputs. Take: \[ x_1=0,\quad x_2=2\pi \] Then: \[ \sin0=0,\quad \sin2\pi=0

Check Function \(f(x)=x-5\) on \( \mathbb{Z} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x-5 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1-5=x_2-5 \] So: \[ x_1=x_2 \] ✔ Hence, function is one-one. 🔹 Step 2: Check

Check Function \(f(x)=x^2+x\) on \( \mathbb{Z} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x^2+x \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) A function is one-one if: \[ f(x_1)=f(x_2)\Rightarrow x_1=x_2 \] Take: \[ x=1,\quad x=-2 \] Then: \[ f(1)=1^2+1=2 \] \[ f(-2)=(-2)^2+(-2)=4-2=2 \]

Check Function \(f(x)=|x|\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=|x| \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) A function is one-one if different inputs have different outputs. Take: \[ x=2,\quad x=-2 \] Then: \[ f(2)=|2|=2,\quad f(-2)=|-2|=2 \] But:

Check Function \(f(x)=x^3\) on \( \mathbb{Z} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Z}\to\mathbb{Z},\quad f(x)=x^3 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3=x_2^3 \] Cube function is strictly increasing on integers. So: \[ x_1=x_2 \] ✔ Hence,

Check Function \(f(x)=x^3\) on \( \mathbb{N} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{N}\to\mathbb{N},\quad f(x)=x^3 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3=x_2^3 \] Cube function is strictly increasing on natural numbers. So: \[ x_1=x_2 \] ✔