Educational

Find \(f\circ g\) 📝 Question Let: \[ f=\{(5,2),(6,3)\}, \quad g=\{(2,5),(3,6)\} \] Find \(f\circ g\). ✅ Solution 🔹 Step 1: Definition \[ (f\circ g)(x)=f(g(x)) \] — 🔹 Step 2: Apply \(g\) first \[ g(2)=5,\quad g(3)=6 \] — 🔹 Step 3: Apply \(f\) \[ f(5)=2,\quad f(6)=3 \] — 🔹 Step 4: Write composition \[ (f\circ g)(2)=2,\quad (f\circ […]

Check Whether the Function is One-One 📝 Question Let: \[ A=\{1,2,3\}, \quad B=\{4,5,6,7\} \] \[ f=\{(1,4),(2,5),(3,6)\} \] Check whether \(f\) is one-one. ✅ Solution 🔹 Step 1: Definition of one-one A function is one-one if different elements of domain have different images. — 🔹 Step 2: Check images \[ f(1)=4,\quad f(2)=5,\quad f(3)=6 \] All outputs

Find \(f(f(x))\) 📝 Question Let: \[ f(x)=3x+2 \] Find \(f(f(x))\). ✅ Solution 🔹 Step 1: Write composition \[ f(f(x))=f(3x+2) \] — 🔹 Step 2: Substitute into function \[ f(f(x))=3(3x+2)+2 \] — 🔹 Step 3: Simplify :contentReference[oaicite:0]{index=0} — 🎯 Final Answer \[ \boxed{f(f(x))=9x+8} \] 🚀 Exam Shortcut Replace \(x\) by \(f(x)\) Simplify step by step Be

Find \(f\circ f(x)\) 📝 Question Let: \[ f(x)=\left(3-x^3\right)^{1/3} \] Find \(f(f(x))\). ✅ Solution 🔹 Step 1: Write composition \[ f(f(x))=\left(3-\left[f(x)\right]^3\right)^{1/3} \] — 🔹 Step 2: Substitute \(f(x)\) \[ f(f(x))=\left(3-\left((3-x^3)^{1/3}\right)^3\right)^{1/3} \] — 🔹 Step 3: Simplify :contentReference[oaicite:0]{index=0} So, \[ f(f(x))=\left(3-(3-x^3)\right)^{1/3} \] \[ =\left(x^3\right)^{1/3} \] — 🔹 Step 4: Final simplification \[ (x^3)^{1/3}=x \] — 🎯 Final

Find Range of \(f(x)=\frac{|x-1|}{x-1}\) 📝 Question Find the range of the function: \[ f(x)=\frac{|x-1|}{x-1} \] ✅ Solution 🔹 Step 1: Domain restriction Denominator cannot be zero: \[ x-1 \ne 0 \Rightarrow x \ne 1 \] — 🔹 Step 2: Consider cases Case 1: \(x>1\) \[ |x-1|=x-1 \Rightarrow f(x)=\frac{x-1}{x-1}=1 \] Case 2: \(x

Find \((f\circ g)(7)\) 📝 Question Let: \[ f(x)=x+7,\quad g(x)=x-7 \] Find \((f\circ g)(7)\). ✅ Solution 🔹 Step 1: Use definition \[ (f\circ g)(x)=f(g(x)) \] — 🔹 Step 2: Find \(g(7)\) \[ g(7)=7-7=0 \] — 🔹 Step 3: Find \(f(g(7))\) \[ f(0)=0+7=7 \] — 🎯 Final Answer \[ \boxed{7} \] 🚀 Exam Shortcut Compute inner function

Determine Nature of \(f(x)=x+\sqrt{x^2}\) 📝 Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x+\sqrt{x^2} \] Determine whether \(f\) is one-one, many-one, onto or into. ✅ Solution 🔹 Step 1: Simplify function We know: :contentReference[oaicite:0]{index=0} So, \[ f(x)=x+|x| \] — 🔹 Step 2: Consider cases Case 1: \(x \ge 0\) \[ f(x)=x+x=2x \] Case 2: \(x < 0\) \[

Find Domain of \(f(x)=\frac{1}{\sqrt{[x]-x}}\) 📝 Question Find the domain of the function: \[ f(x)=\frac{1}{\sqrt{[x]-x}} \] where \([x]\) denotes the greatest integer function. ✅ Solution 🔹 Step 1: Conditions for denominator For the function to be defined: \([x]-x \ge 0\) (inside square root) \([x]-x \ne 0\) (denominator cannot be zero) So, \[ [x]-x > 0 \]

Find Domain of \(f(x)=\sqrt{[x]-x}\) 📝 Question Find the domain of the function: \[ f(x)=\sqrt{[x]-x} \] where \([x]\) denotes the greatest integer function. ✅ Solution 🔹 Step 1: Condition for square root For the function to be defined: \[ [x]-x \ge 0 \] — 🔹 Step 2: Use property of greatest integer function We know: \[

Find Domain of \(f(x)=\sqrt{x}-[x]\) 📝 Question Find the domain of the function: \[ f(x)=\sqrt{x}-[x] \] where \([x]\) denotes the greatest integer function. ✅ Solution 🔹 Step 1: Condition for \(\sqrt{x}\) For square root to be defined: \[ x \ge 0 \] — 🔹 Step 2: Condition for \([x]\) The greatest integer function is defined for