Inverse Hyperbolic Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to (-1,1) \) be defined by \[ f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}} \] Find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Recognize Form \[ f(x)=\tanh x \] — 🔹 Step 2: Let \(y=f(x)\) \[ y=\frac{e^x-e^{-x}}{e^x+e^{-x}} \] — 🔹 Step 3: Solve for \(x\) Multiply: \[ y(e^x+e^{-x})=e^x-e^{-x} \] […]
Find f and g Find Functions \(f\) and \(g\) 🎥 Video Explanation 📝 Question Given: \[ g(f(x)) = |\sin x| \] \[ f(g(x)) = (\sin \sqrt{x})^2 \] A. \(f(x)=\sin^2 x,\; g(x)=\sqrt{x}\) B. \(f(x)=\sin x,\; g(x)=|x|\) C. \(f(x)=x^2,\; g(x)=\sin \sqrt{x}\) D. cannot be determined ✅ Solution 🔹 Step 1: Analyze \(f(g(x))\) \[ f(g(x)) = (\sin \sqrt{x})^2
Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=3x-5 \] A. \(\frac{1}{3x-5}\) B. \(\frac{x+5}{3}\) C. does not exist (not one-one) D. does not exist (not onto) ✅ Solution 🔹 Step 1: Check Invertibility Linear function with non-zero slope ⇒ one-one and onto. ✔️ Inverse exists — 🔹
Function Composition Equation Solve \(f \circ g(x) = g \circ f(x)\) 🎥 Video Explanation 📝 Question Given: \[ f(x)=x^2,\quad g(x)=2^x \] Find solution of: \[ f(g(x)) = g(f(x)) \] A. \(\mathbb{R}\) B. \(\{0\}\) C. \(\{0,2\}\) D. none of these ✅ Solution 🔹 Step 1: Compute Both Sides \[ f(g(x)) = f(2^x) = (2^x)^2 = 2^{2x}
Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=6^x+6^{|x|} \] 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 Form Case 1: \(x \ge 0\) \[ f(x)=6^x + 6^x = 2\cdot
Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\mathbb{Z} \to \mathbb{Z} \) be defined by: \[ f(x)= \begin{cases} \dfrac{x}{2}, & \text{if } x \text{ is even} \\ 0, & \text{if } x \text{ is odd} \end{cases} \] A. onto but not one-one B. one-one but not onto C. one-one
Check Bijective Functions Check Which Functions are Bijective 🎥 Video Explanation 📝 Question Let \(A=\{x\in\mathbb{R}:-1\le x\le 1\}\). Which of the following functions \(A \to A\) are bijections? A. \(f(x)=|x|\) B. \(f(x)=\sin\left(\frac{\pi x}{2}\right)\) C. \(f(x)=\sin\left(\frac{\pi x}{4}\right)\) D. none of these ✅ Solution 🔹 Option A: \(f(x)=|x|\) \[ f(-x)=f(x) \] ❌ Not one-one ⇒ Not bijective —
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
Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=x^2 \] A. injective but not surjective B. surjective but not injective C. both injective and surjective D. neither injective nor surjective ✅ Solution 🔹 Step 1: Check Injective \[ f(x)=x^2 \Rightarrow f(2)=4,\; f(-2)=4 \] Different
Check Function Type Check Injective / Surjective 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \) be defined by \[ f(x)=\frac{e^{x^2}-e^{-x^2}}{e^{x^2}+e^{-x^2}} \] A. one-one but not onto B. one-one and onto C. onto but not one-one D. neither one-one nor onto ✅ Solution 🔹 Step 1: Simplify \[ f(x)=\tanh(x^2) \] — 🔹 Step