Ravi Kant Kumar

Find \(f \circ g\) and \(g \circ f\) for \(f(x)=\sin^{-1}x\) and \(g(x)=x^2\) 📺 Video Explanation 📝 Question Let functions be defined as: \[ f(x)=\sin^{-1}x,\qquad g(x)=x^2 \] Find: \((f\circ g)(x)\) \((g\circ f)(x)\) ✅ Solution 🔹 Find \((f\circ g)(x)\) By definition: \[ (f\circ g)(x)=f(g(x)) \] Substitute \(g(x)=x^2\): \[ (f\circ g)(x)=\sin^{-1}(x^2) \] For \(\sin^{-1}(x^2)\) to be defined: \[ […]

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

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

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

Find \(f \circ g\) and \(g \circ f\) for \(f(x)=e^x\) and \(g(x)=\ln x\) 📺 Video Explanation 📝 Question Let functions \(f:\mathbb{R}\to\mathbb{R}\) and \(g:(0,\infty)\to\mathbb{R}\) be defined as: \[ f(x)=e^x,\qquad g(x)=\ln x \] Find: \((f\circ g)(x)\) \((g\circ f)(x)\) ✅ Solution 🔹 Find \((f\circ g)(x)\) By definition: \[ (f\circ g)(x)=f(g(x)) \] Substitute: \[ (f\circ g)(x)=f(\ln x) \] Since:

Show \(g \circ f\) is Onto if \(f\) and \(g\) are Onto Functions 📺 Video Explanation 📝 Question If: \[ f:A\to B \quad \text{and} \quad g:B\to C \] are onto (surjective) functions, show that: \[ g\circ f:A\to C \] is also an onto function. ✅ Solution 🔹 Step 1: Use definition of onto function A

Show \(g \circ f\) is One-One if \(f\) and \(g\) are One-One Functions 📺 Video Explanation 📝 Question If: \[ f:A\to B \quad \text{and} \quad g:B\to C \] are one-one (injective) functions, show that: \[ g\circ f:A\to C \] is also a one-one function. ✅ Solution 🔹 Step 1: Use definition of one-one function A

Example Where \(g \circ f\) is Injective but \(g\) is Not Injective 📺 Video Explanation 📝 Question Give examples of two functions: \[ f:\mathbb{N}\to\mathbb{Z},\qquad g:\mathbb{Z}\to\mathbb{Z} \] such that: \[ g\circ f \text{ is injective, but } g \text{ is not injective.} \] ✅ Solution 🔹 Choose Functions Take: \[ f(x)=x \] and \[ g(x)=|x| \]

Example Where \(g \circ f\) is Onto but \(f\) is Not Onto 📺 Video Explanation 📝 Question Give examples of two functions: \[ f:\mathbb{N}\to\mathbb{N},\qquad g:\mathbb{N}\to\mathbb{N} \] such that: \[ g\circ f \text{ is onto, but } f \text{ is not onto.} \] ✅ Solution 🔹 Choose Functions Take: \[ f(x)=x+1 \] and \[ g(x)= \begin{cases}

Verify \(h\circ(g\circ f)=(h\circ g)\circ f\) for Given Functions 📺 Video Explanation 📝 Question Let: \[ f:\mathbb{N}\to\mathbb{N},\qquad f(x)=2x \] \[ g:\mathbb{N}\to\mathbb{N},\qquad g(y)=3y+4 \] \[ h:\mathbb{N}\to\mathbb{R},\qquad h(z)=\sin z \] Show that: \[ h\circ(g\circ f)=(h\circ g)\circ f \] ✅ Solution 🔹 Step 1: Find \(g\circ f\) By definition: \[ (g\circ f)(x)=g(f(x)) \] Substitute \(f(x)=2x\): \[ (g\circ f)(x)=g(2x)=3(2x)+4 \]