Educational

Number of Surjections Find Number of Surjections 🎥 Video Explanation 📝 Question Let \(A=\{1,2,\dots,n\}\) and \(B=\{a,b\}\). Find number of surjections from \(A\) to \(B\). (a) \(nP_2\) (b) \(2^n – 2\) (c) \(2^n – 1\) (d) \(nC_2\) ✅ Solution 🔹 Step 1: Total Functions Each element of \(A\) can map to either \(a\) or \(b\): \[ […]

Piecewise Function Evaluate \(f(-1)+f(2)+f(4)\) 🎥 Video Explanation 📝 Question Let: \[ f(x)= \begin{cases} 2x, & x>3 \\ x^2, & 1

Inverse Trigonometric Function Find \(f^{-1}(1)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=\tan x \] Find \(f^{-1}(1)\). (a) \(\frac{\pi}{4}\) (b) \(\{n\pi+\frac{\pi}{4}: n\in\mathbb{Z}\}\) (c) does not exist (d) none of these ✅ Solution 🔹 Step 1: Check Invertibility \(\tan x\) is periodic: \[ \tan x = \tan(x+n\pi) \] So it is

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=x^2-3 \] (a) \(\sqrt{x+3}\) (b) \(-\sqrt{x+3}\) (c) \(x+\sqrt{3}\) (d) none of these ✅ Solution 🔹 Step 1: Check Injectivity \[ f(x)=x^2-3 \Rightarrow f(2)=1,\; f(-2)=1 \] Different inputs → same output ⇒ ❌ Not one-one — 🔹 Step 2: Conclusion

Domain of Inverse Function Find Domain of \(f^{-1}\) 🎥 Video Explanation 📝 Question Let: \[ f(x)=x^3 \] Domain of \(f\): \(\{0,1,2,3\}\) Find domain of \(f^{-1}\). (a) \(\{3,2,1,0\}\) (b) \(\{0,-1,-2,-3\}\) (c) \(\{0,1,8,27\}\) (d) \(\{0,-1,-8,-27\}\) ✅ Solution 🔹 Step 1: Find Range of \(f\) \[ f(0)=0,\quad f(1)=1,\quad f(2)=8,\quad f(3)=27 \] Range: \[ \{0,1,8,27\} \] — 🔹 Step

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=x^3+3 \] (a) \(x^{1/3}-3\) (b) \(x^{1/3}+3\) (c) \((x-3)^{1/3}\) (d) \((x+3)^{1/3}\) ✅ Solution 🔹 Step 1: Let \(y=f(x)\) \[ y=x^3+3 \] — 🔹 Step 2: Solve for \(x\) \[ x^3=y-3 \] \[ x=(y-3)^{1/3} \] — 🔹 Step 3: Replace \(y\)

Find g(x) Find \(g(x)\) 🎥 Video Explanation 📝 Question Given: \[ f(x)=\sin^2 x \] \[ g(f(x))=|\sin x| \] Find \(g(x)\). (a) \(\sqrt{x-1}\) (b) \(\sqrt{x}\) (c) \(\sqrt{x+1}\) (d) \(-\sqrt{x}\) ✅ Solution 🔹 Step 1: Substitute \(f(x)\) \[ g(\sin^2 x)=|\sin x| \] — 🔹 Step 2: Express RHS \[ |\sin x|=\sqrt{\sin^2 x} \] — 🔹 Step 3:

Find f(x) Find \(f(x)\) 🎥 Video Explanation 📝 Question Given: \[ g(x)=x^2+x-2 \] \[ \frac{1}{2}g(f(x))=2x^2-5x+2 \] Find \(f(x)\). ✅ Solution 🔹 Step 1: Remove Fraction \[ g(f(x))=4x^2-10x+4 \] — 🔹 Step 2: Substitute \(g(x)\) \[ (f(x))^2 + f(x) -2 = 4x^2-10x+4 \] — 🔹 Step 3: Rearrange \[ (f(x))^2 + f(x) – (4x^2-10x+6)=0 \] —

Function Composition Evaluate \(f(g(h(x)))\) 🎥 Video Explanation 📝 Question Let: \[ f(x)=\sin^{-1}x,\quad g(x)=[x^2],\quad h(x)=2x \] where \([x]\) denotes greatest integer ≤ \(x\), and \[ \frac{1}{2} \le x \le \frac{1}{\sqrt{2}} \] Find \(f(g(h(x)))\). ✅ Solution 🔹 Step 1: Find \(h(x)\) \[ h(x)=2x \] \[ 1 \le 2x \le \sqrt{2} \] — 🔹 Step 2: Find \(g(h(x))\)

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question \[ f(x)=\frac{-x|x|}{1+x^2}, \quad f:\mathbb{R}\to(-1,1) \] Find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Case-wise Expression For \(x \ge 0\): \[ f(x)=\frac{-x^2}{1+x^2} \] For \(x < 0\): \[ f(x)=\frac{x^2}{1+x^2} \] — 🔹 Step 2: Let \(y=f(x)\) Case 1: \[ y=\frac{x^2}{1+x^2} \Rightarrow x^2=\frac{y}{1-y} \] Case 2: \[ y=\frac{-x^2}{1+x^2} \Rightarrow