April 2026

Show \(f\) and \(g\) are Bijective and Verify \((g \circ f)^{-1}=f^{-1}\circ g^{-1}\) 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{Q}\to\mathbb{Q},\qquad f(x)=2x \] and: \[ g:\mathbb{Q}\to\mathbb{Q},\qquad g(x)=x+2 \] Show that both are bijections and verify: \[ (g\circ f)^{-1}=f^{-1}\circ g^{-1} \] ✅ Solution 🔹 Step 1: Show that \(f(x)=2x\) is bijective One-one: If: \[ f(x_1)=f(x_2) \] Then: […]

Check Whether \(f(x)=x^3+4\) is Bijective and Find \(f^{-1}(3)\) 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R},\qquad f(x)=x^3+4 \] Check whether \(f\) is bijection. If yes, find: \[ f^{-1}(3) \] ✅ Solution 🔹 Step 1: Check one-one Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3+4=x_2^3+4 \] So: \[ x_1^3=x_2^3 \] Hence: \[ x_1=x_2 \] Therefore: \[ f

Show \(f(x)=x^3-3\) is Invertible and Find \(f^{-1}\) 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R},\qquad f(x)=x^3-3 \] Prove that inverse exists and find: \[ f^{-1}(x) \] Also find: \[ f^{-1}(24),\qquad f^{-1}(5) \] ✅ Solution 🔹 Step 1: Show that \(f\) is one-one Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3-3=x_2^3-3 \] So: \[ x_1^3=x_2^3 \] Taking

Show \(f(x)=9x^2+6x-5\) is Invertible on \(\mathbb{R}_+\) and Find \(f^{-1}\) 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{R}_+\to[-5,\infty),\qquad f(x)=9x^2+6x-5 \] where \(\mathbb{R}_+\) denotes the set of all non-negative real numbers. Show that \(f\) is invertible and find: \[ f^{-1}(x)=\frac{\sqrt{x+6}-1}{3} \] ✅ Solution 🔹 Step 1: Show that \(f\) is one-one Take: \[ x_1,x_2\in\mathbb{R}_+ \] and assume:

Show \(f \circ f(x)=x\) and Find the Inverse of \(f(x)=\frac{4x+3}{6x-4}\) 📺 Video Explanation 📝 Question Let: \[ f(x)=\frac{4x+3}{6x-4},\qquad x\ne\frac{2}{3} \] Show that: \[ (f\circ f)(x)=x \] for all: \[ x\ne\frac{2}{3} \] Also find: \[ f^{-1}(x) \] ✅ Solution 🔹 Step 1: Find \(f(f(x))\) By definition: \[ (f\circ f)(x)=f\left(\frac{4x+3}{6x-4}\right) \] Put: \[ t=\frac{4x+3}{6x-4} \] Then: \[

Show \(f(x)=x^2+4\) is Invertible on \(\mathbb{R}_+\) and Find \(f^{-1}\) 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{R}_+\to[4,\infty),\qquad f(x)=x^2+4 \] where \(\mathbb{R}_+\) is the set of all non-negative real numbers. Show that \(f\) is invertible and: \[ f^{-1}(x)=\sqrt{x-4} \] Also verify: \[ f^{-1}\circ f(x)=x \] ✅ Solution 🔹 Step 1: Show that \(f\) is one-one Let:

Show \(f(x)=4x+3\) is Invertible on \(\mathbb{R}\) and Find \(f^{-1}\) 📺 Video Explanation 📝 Question Show that: \[ f:\mathbb{R}\to\mathbb{R},\qquad f(x)=4x+3 \] is invertible. Find: \[ f^{-1}(x) \] ✅ Solution 🔹 Step 1: Show that \(f\) is one-one Assume: \[ f(x_1)=f(x_2) \] Then: \[ 4x_1+3=4x_2+3 \] Subtract 3: \[ 4x_1=4x_2 \] Divide by 4: \[ x_1=x_2 \]

Show \(f(x)=3x+5\) is Invertible on \(\mathbb{Q}\) and Find \(f^{-1}\) 📺 Video Explanation 📝 Question Show that: \[ f:\mathbb{Q}\to\mathbb{Q},\qquad f(x)=3x+5 \] is invertible. Also find: \[ f^{-1}(x) \] ✅ Solution 🔹 Step 1: Show that \(f\) is one-one Assume: \[ f(x_1)=f(x_2) \] Then: \[ 3x_1+5=3x_2+5 \] Subtract 5: \[ 3x_1=3x_2 \] Divide by 3: \[ x_1=x_2

Find \((g \circ f)^{-1}\) and Verify \((g \circ f)^{-1}=f^{-1}\circ g^{-1}\) 📺 Video Explanation 📝 Question Let: \[ A=\{1,2,3,4\},\quad B=\{3,5,7,9\},\quad C=\{7,23,47,79\} \] and: \[ f:A\to B,\qquad f(x)=2x+1 \] \[ g:B\to C,\qquad g(x)=x^2-2 \] Find: \[ (g\circ f)^{-1}\quad \text{and}\quad f^{-1}\circ g^{-1} \] as sets of ordered pairs, and verify: \[ (g\circ f)^{-1}=f^{-1}\circ g^{-1} \] ✅ Solution 🔹

Show \(f\), \(g\), and \(g \circ f\) are Invertible and Verify Inverse Rule 📺 Video Explanation 📝 Question Let: \[ f:\{1,2,3\}\to\{a,b,c\} \] defined by: \[ f(1)=a,\quad f(2)=b,\quad f(3)=c \] and: \[ g:\{a,b,c\}\to\{\text{apple, ball, cat}\} \] defined by: \[ g(a)=\text{apple},\quad g(b)=\text{ball},\quad g(c)=\text{cat} \] Show that \(f\), \(g\), and \(g\circ f\) are invertible. Find: \(f^{-1}\) \(g^{-1}\) \((g\circ