April 2026

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, \[ […]

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

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: Understand Meaning of \(f^{-1}(25)\) Since \(f(x)=x^2\) is not one-one on \(\mathbb{R}\), inverse function does not exist. Here, \(f^{-1}(25)\) means inverse image of 25. — 🔹 Step 2: Solve Equation \[ f(x)=25 \] \[ x^2=25 \] \[

Find Number of One-One Functions from \(A\) to \(B\) 📝 Question Let: \[ A=\{1,2,3,4\}, \quad B=\{a,b,c\} \] Find the total number of one-one (injective) functions from \(A\) to \(B\). ✅ Solution 🔹 Step 1: Compare Sizes \[ |A|=4,\quad |B|=3 \] For a function to be one-one, each element of \(A\) must map to a distinct

Find Number of One-One Functions from \(A\) to \(B\) 📝 Question Let: \[ A=\{a,b,c\}, \quad B=\{-2,-1,0,1,2\} \] Find the total number of one-one (injective) functions from \(A\) to \(B\). ✅ Solution 🔹 Step 1: Use Formula If \(|A|=m\) and \(|B|=n\) with \(n \ge m\), then number of one-one functions is: \[ {}^{n}P_{m}=\frac{n!}{(n-m)!} \] (Permutation formula)

Find Total Number of Functions from \(A\) to \(B\) 📝 Question Let: \[ A=\{1,2,3\}, \quad B=\{a,b\} \] Find the total number of functions from \(A\) to \(B\). ✅ Solution 🔹 Step 1: Use Formula If a set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, then: \[ \text{Number of functions} = n^m \]

Which of the Following Graphs Represent a One-One Function? 📝 Question Determine which of the following graphs represents a one-one function. (a) and (b) are given graphs. ✅ Solution 🔹 Step 1: Use Horizontal Line Test A function is one-one if every horizontal line intersects the graph at most once. 🔹 Step 2: Analyze Graph

Prove that \(f\circ g\) is Injection and Surjection 📝 Question Let \(f:A\to A\) and \(g:A\to A\) be two bijections. Prove that: (i) \(f\circ g\) is injective (ii) \(f\circ g\) is surjective ✅ Solution 🔹 Step 1: Given Since \(f\) and \(g\) are bijections, they are both: Injective (one-one) Surjective (onto) — 🔹 Step 2: Prove

Which of the Following Graphs Represent a Function? 📝 Question Determine which of the following graphs represents a function. (a) and (b) are given graphs. ✅ Solution 🔹 Step 1: Use Vertical Line Test A graph represents a function if every vertical line intersects the graph at most once. 🔹 Step 2: Analyze Graph (a)

Prove Existence of Bijection from Injective Maps 📝 Question Let \(A\) and \(B\) be two finite sets. Suppose there exists: \[ f:A\to B \quad \text{(injective)} \] \[ g:B\to A \quad \text{(injective)} \] Prove that there exists a bijection from \(A\) to \(B\). ✅ Solution 🔹 Step 1: Use injectivity to compare sizes Since \(f:A\to B\)