Educational

Find \(g \circ f\) and \(f \circ g\) for \(f(x)=x\) and \(g(x)=|x|\) 📺 Video Explanation 📝 Question Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as: \[ f(x)=x,\qquad g(x)=|x| \] Find: \((g\circ f)(x)\) \((f\circ g)(x)\) ✅ Solution 🔹 Find \((g\circ f)(x)\) By definition: \[ (g\circ f)(x)=g(f(x)) \] Since: \[ f(x)=x \] Substitute: \[ g(f(x))=g(x) \] Now: […]

Find \(g \circ f\) and \(f \circ g\) for \(f(x)=x^2+8\) and \(g(x)=3x^3+1\) 📺 Video Explanation 📝 Question Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as: \[ f(x)=x^2+8,\qquad g(x)=3x^3+1 \] Find: \((g\circ f)(x)\) \((f\circ g)(x)\) ✅ Solution 🔹 Find \((g\circ f)(x)\) By definition: \[ (g\circ f)(x)=g(f(x)) \] Substitute \(f(x)=x^2+8\): \[ g(f(x))=g(x^2+8) \] Since: \[ g(x)=3x^3+1 \]

Find \(g \circ f\) and \(f \circ g\) for \(f(x)=2x+x^2\) and \(g(x)=x^3\) 📺 Video Explanation 📝 Question Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as: \[ f(x)=2x+x^2,\qquad g(x)=x^3 \] Find: \((g\circ f)(x)\) \((f\circ g)(x)\) ✅ Solution 🔹 Find \((g\circ f)(x)\) By definition: \[ (g\circ f)(x)=g(f(x)) \] Substitute \(f(x)=2x+x^2\): \[ g(f(x))=g(2x+x^2) \] Since: \[ g(x)=x^3 \]

Find \(g \circ f\) and \(f \circ g\) for \(f(x)=2x+3\) and \(g(x)=x^2+5\) 📺 Video Explanation 📝 Question Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}\to\mathbb{R}\) be defined as: \[ f(x)=2x+3,\qquad g(x)=x^2+5 \] Find: \((g\circ f)(x)\) \((f\circ g)(x)\) ✅ Solution 🔹 Find \((g\circ f)(x)\) By definition: \[ (g\circ f)(x)=g(f(x)) \] Substitute \(f(x)=2x+3\): \[ g(f(x))=g(2x+3) \] Since: \[ g(x)=x^2+5 \]

Prove Piecewise Function is a Bijection 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{N}\to\mathbb{N} \] be defined by: \[ f(n)= \begin{cases} n+1,& \text{if } n \text{ is odd}\\ n-1,& \text{if } n \text{ is even} \end{cases} \] Show that \(f\) is bijective. ✅ Solution 🔹 Understand the Mapping Function swaps consecutive numbers: \(1\to2\) \(2\to1\) \(3\to4\)

Show \(f(x)=x-[x]\) is Neither One-One Nor Onto 📺 Video Explanation 📝 Question Show that: \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=x-[x] \] is neither one-one nor onto. ✅ Solution 🔹 Step 1: Understand Function Here: \[ [x] \] means greatest integer less than or equal to \(x\). So: \[ f(x)=x-[x] \] is fractional part of \(x\). Thus: \[ 0\leq

Construct Mappings from \(A\) to \(B\) 📺 Video Explanation 📝 Question Given: \[ A=\{2,3,4\},\quad B=\{2,5,6,7\} \] Construct examples of: (i) an injective map from \(A\) to \(B\) (ii) a mapping from \(A\) to \(B\) which is not injective (iii) a mapping from \(A\) to \(B\) ✅ Solution 🔹 (i) Injective Map Choose distinct images for

Product of Two One-One Functions Need Not Be One-One 📺 Video Explanation 📝 Question Show that if: \[ f_1,f_2:\mathbb{R}\to\mathbb{R} \] are one-one, then: \[ (f_1\times f_2)(x)=f_1(x)f_2(x) \] need not be one-one. ✅ Solution Take: \[ f_1(x)=x \] and: \[ f_2(x)=\frac{1}{x},\quad x\neq0 \] (Or restrict domain excluding 0.) 🔹 Check \(f_1\) Function: \[ f_1(x)=x \] is

Is Quotient of Two One-One Functions Always One-One? 📺 Video Explanation 📝 Question Suppose: \[ f_1,f_2:\mathbb{R}\to\mathbb{R} \] are non-zero one-one functions. Is: \[ \left(\frac{f_1}{f_2}\right)(x)=\frac{f_1(x)}{f_2(x)} \] necessarily one-one? ✅ Solution No, quotient need not be one-one. 🔹 Counter Example Take: \[ f_1(x)=x,\quad f_2(x)=x^2 \] on: \[ \mathbb{R}\setminus\{0\} \] Both are non-zero and one-one on suitable restricted

Examples of Two Surjective Functions Whose Sum is Not Surjective 📺 Video Explanation 📝 Question Give examples of two surjective functions: \[ f_1,f_2:\mathbb{Z}\to\mathbb{Z} \] such that: \[ (f_1+f_2)(x)=f_1(x)+f_2(x) \] is not surjective. ✅ Solution Take: \[ f_1(x)=x \] and: \[ f_2(x)=x \] 🔹 Check \(f_1\) For any: \[ y\in\mathbb{Z} \] choose: \[ x=y \] Then: