Ravi Kant Kumar

Find \(f^{-1}(x)\) for \(f(x)=3x-4\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=3x-4 \] Given that \(f\) is invertible, find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Check invertibility The function \(f(x)=3x-4\) is linear with non-zero slope. Hence, it is one-one and onto, so inverse exists. — 🔹 Step 2: Find inverse Let: \[ y=3x-4 \] Interchange \(x\) […]

Find Range of \(f(x)=\cos([x])\) 📝 Question Let: \[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to \mathbb{R}, \quad f(x)=\cos([x]) \] where \([x]\) denotes the greatest integer function. Find the range of \(f\). ✅ Solution 🔹 Step 1: Find possible values of \([x]\) \[ -\frac{\pi}{2} \approx -1.57,\quad \frac{\pi}{2} \approx 1.57 \] So \(x \in (-1.57, 1.57)\) Thus possible integer values of \([x]\) are:

Find \(f^{-1}(x)\) for \(f(x)=10x-7\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=10x-7 \] Find \(f^{-1}(x)\). ✅ Solution 🔹 Step 1: Check invertibility The function \(f(x)=10x-7\) is linear with non-zero slope. Hence, it is one-one and onto, so inverse exists. — 🔹 Step 2: Find inverse Let: \[ y=10x-7 \] Interchange \(x\) and \(y\): \[ x=10y-7 \]

Find \(f^{-1}(-1)\) for \(f(x)=(x-2)^3\) on \(\mathbb{C}\) 📝 Question Let: \[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=(x-2)^3 \] Find \(f^{-1}(-1)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(-1)\) Here, \(f^{-1}(-1)\) means the inverse image of \(-1\). — 🔹 Step 2: Solve Equation \[ f(x)=-1 \] \[ (x-2)^3=-1 \] — 🔹 Step 3: Find Cube Roots Let \(y=x-2\). Then: \[

Find \(f^{-1}(-25)\) for \(f(x)=x^2\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x^2 \] Find \(f^{-1}(-25)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(-25)\) Since \(f(x)=x^2\) is not one-one, inverse function does not exist. Here, \(f^{-1}(-25)\) means the inverse image of \(-25\). — 🔹 Step 2: Solve Equation \[ f(x)=-25 \] \[ x^2=-25 \] — 🔹 Step

Find \(f^{-1}(1)\) for \(f(x)=x^4\) on \(\mathbb{C}\) 📝 Question Let: \[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^4 \] Find \(f^{-1}(1)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(1)\) Since \(f(x)=x^4\) is not one-one on \(\mathbb{C}\), inverse function does not exist. Here, \(f^{-1}(1)\) means the inverse image of 1. — 🔹 Step 2: Solve Equation \[ f(x)=1 \] \[ x^4=1

Find \(f^{-1}(1)\) for \(f(x)=x^4\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x^4 \] Find \(f^{-1}(1)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(1)\) The function \(f(x)=x^4\) is not one-one on \(\mathbb{R}\). So, \(f^{-1}(1)\) means the inverse image of 1. — 🔹 Step 2: Solve Equation \[ f(x)=1 \] \[ x^4=1 \] — 🔹 Step 3: Find

Find \(f^{-1}(-1)\) for \(f(x)=x^3\) on \(\mathbb{C}\) 📝 Question Let: \[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^3 \] Find \(f^{-1}(-1)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(-1)\) Since \(f(x)=x^3\) is not one-one on \(\mathbb{C}\), inverse function does not exist. Here, \(f^{-1}(-1)\) means the inverse image of \(-1\). — 🔹 Step 2: Solve Equation \[ f(x)=-1 \] \[ x^3=-1

Find \(f^{-1}(1)\) for \(f(x)=x^3\) on \(\mathbb{C}\) 📝 Question Let: \[ f:\mathbb{C}\to\mathbb{C}, \quad f(x)=x^3 \] Find \(f^{-1}(1)\). ✅ Solution 🔹 Step 1: Meaning of \(f^{-1}(1)\) Since \(f(x)=x^3\) is not one-one on \(\mathbb{C}\), inverse function does not exist. Here, \(f^{-1}(1)\) means the inverse image of 1. — 🔹 Step 2: Solve Equation \[ f(x)=1 \] \[ x^3=1

Find \(f^{-1}(1)\) for \(f(x)=x^3\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x^3 \] Find \(f^{-1}(1)\). ✅ Solution 🔹 Step 1: Check invertibility The function \(f(x)=x^3\) is strictly increasing on \(\mathbb{R}\). Hence, it is one-one and onto, so inverse exists. — 🔹 Step 2: Find inverse function Let: \[ y=x^3 \] Taking cube root: :contentReference[oaicite:0]{index=0} Thus, \[