Educational

Invertible Function Find Set \(X\) for Invertibility 🎥 Video Explanation 📝 Question Let \( f:[2,\infty)\to X \), \[ f(x)=4x-x^2 \] (a) \([2,\infty)\) (b) \((-\infty,2]\) (c) \((-\infty,4]\) (d) \([4,\infty)\) ✅ Solution 🔹 Step 1: Rewrite \[ f(x)=4x-x^2 \] \[ =-(x^2-4x) =-(x-2)^2+4 \] — 🔹 Step 2: Domain Analysis Given domain: \([2,\infty)\) Function is decreasing on this […]

Linear Function Mapping Find Linear Functions Mapping Intervals 🎥 Video Explanation 📝 Question Find all linear functions mapping \([-1,1]\) onto \([0,2]\). A. \(f(x)=x+1,\; g(x)=-x+1\) B. \(f(x)=x-1,\; g(x)=x+1\) C. \(f(x)=-x-1,\; g(x)=x-1\) D. none of these ✅ Solution 🔹 Step 1: General Form Let: \[ f(x)=ax+b \] — 🔹 Step 2: Endpoint Mapping Case 1: Increasing function

Function Composition Find \(\alpha\) such that \(f(f(x))=x\) 🎥 Video Explanation 📝 Question Given: \[ f(x)=\frac{\alpha x}{x+1}, \quad x\ne -1 \] Find \(\alpha\) such that: \[ f(f(x))=x \] (a) \(\sqrt{2}\) (b) \(-\sqrt{2}\) (c) \(1\) (d) \(-1\) ✅ Solution 🔹 Step 1: Compute \(f(f(x))\) \[ f(f(x)) = f\!\left(\frac{\alpha x}{x+1}\right) \] \[ = \frac{\alpha \cdot \frac{\alpha x}{x+1}}{\frac{\alpha x}{x+1}+1}

Function Composition Evaluate \(f(g(x))\) 🎥 Video Explanation 📝 Question Given: \[ g(x)=1+x-[x] \] \[ f(x)= \begin{cases} -1, & x0 \end{cases} \] (a) \(x\) (b) \(1\) (c) \(f(x)\) (d) \(g(x)\) ✅ Solution 🔹 Step 1: Understand \(g(x)\) \[ x-[x] \in [0,1) \] So: \[ g(x)=1+(x-[x]) \in [1,2) \] — 🔹 Step 2: Apply \(f\) Since \(g(x)

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:[1,\infty) \to [2,\infty) \), \[ f(x)=x+\frac{1}{x} \] Find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Let \(y=f(x)\) \[ y=x+\frac{1}{x} \] — 🔹 Step 2: Multiply \[ yx=x^2+1 \] \[ x^2-yx+1=0 \] — 🔹 Step 3: Solve Quadratic \[ x=\frac{y\pm\sqrt{y^2-4}}{2} \] — 🔹 Step 4: Choose

Inverse of Greatest Integer Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \( f:\mathbb{R} \to \mathbb{R} \), \[ f(x)=x-[x] \] Find \(f^{-1}(x)\). (a) \(\frac{1}{x-[x]}\) (b) \([x]-x\) (c) not defined (d) none of these ✅ Solution 🔹 Step 1: Understand Function \[ f(x)=x-[x] \] This is the fractional part of \(x\). \[ f(x)\in[0,1) \] —

Function Composition Find \(f\circ(f\circ f)(x)\) 🎥 Video Explanation 📝 Question Given: \[ f(x)=\frac{1}{1-x} \] Find: \[ f(f(f(x))) \] A. \(x\) for all \(x\in\mathbb{R}\) B. \(x\) for all \(x\in\mathbb{R}\setminus\{1\}\) C. \(x\) for all \(x\in\mathbb{R}\setminus\{0,1\}\) D. none of these ✅ Solution 🔹 Step 1: Compute \(f(f(x))\) \[ f(f(x)) = f\!\left(\frac{1}{1-x}\right) \] \[ = \frac{1}{1-\frac{1}{1-x}} \] \[ =

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \(A=\{x\in\mathbb{R}:x\le1\}\) \[ f(x)=x(2-x) \] (a) \(1+\sqrt{1-x}\) (b) \(1-\sqrt{1-x}\) (c) \(\sqrt{1-x}\) (d) \(1\pm\sqrt{1-x}\) ✅ Solution 🔹 Step 1: Rewrite Function \[ f(x)=2x-x^2 \] \[ =1-(x-1)^2 \] — 🔹 Step 2: Let \(y=f(x)\) \[ y=1-(x-1)^2 \] — 🔹 Step 3: Solve for \(x\) \[ (x-1)^2=1-y \] \[

Inverse Function Find \(f^{-1}(x)\) 🎥 Video Explanation 📝 Question Let \(A=\{x\in\mathbb{R}:x\ge1\}\). \[ f(x)=2^x(x-1) \] Find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Let \(y=f(x)\) \[ y=2^x(x-1) \] — 🔹 Step 2: Try Substitution Let \(t=x-1\) ⇒ \(x=t+1\) \[ y=2^{t+1}\cdot t = 2\cdot 2^t \cdot t \] \[ \frac{y}{2}=t\cdot 2^t \] — 🔹 Step 3: Recognize Form

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: \[ y(e^x+e^{-x})=e^x-e^{-x} \]