Check Function \(h(x)=x^2\) on \([-1,1]\) 📺 Video Explanation 📝 Question Let: \[ A=[-1,1] \] Discuss whether the function: \[ h:A\to A,\quad h(x)=x^2 \] is one-one, onto, or bijective. ✅ Solution 🔹 Check One-One (Injective) Take: \[ x=1,\quad x=-1 \] Then: \[ h(1)=1,\quad h(-1)=1 \] Different inputs give same output. ❌ Not one-one. 🔹 Check Onto […]
Check Function \(g(x)=|x|\) on \([-1,1]\) 📺 Video Explanation 📝 Question Let: \[ A=[-1,1] \] Discuss whether the function: \[ g:A\to A,\quad g(x)=|x| \] is one-one, onto, or bijective. ✅ Solution 🔹 Check One-One (Injective) Take: \[ x=1,\quad x=-1 \] Then: \[ g(1)=1,\quad g(-1)=1 \] Different inputs give same output. ❌ Not one-one. 🔹 Check Onto
Check Function \(f(x)=\dfrac{x}{2}\) on \([-1,1]\) 📺 Video Explanation 📝 Question Let: \[ A=[-1,1] \] Discuss whether the function: \[ f:A\to A,\quad f(x)=\frac{x}{2} \] is one-one, onto, or bijective. ✅ Solution 🔹 Check One-One (Injective) Assume: \[ f(x_1)=f(x_2) \] Then: \[ \frac{x_1}{2}=\frac{x_2}{2} \] So: \[ x_1=x_2 \] ✔ Function is one-one. 🔹 Check Onto (Surjective) Since:
Prove \(f(x)=\dfrac{x-2}{x-3}\) is a Bijection 📺 Video Explanation 📝 Question Show that: \[ f:\mathbb{R}\setminus\{3\}\to\mathbb{R}\setminus\{1\} \] defined by: \[ f(x)=\frac{x-2}{x-3} \] is a bijection. ✅ Solution 🔹 Step 1: Prove One-One (Injective) Assume: \[ f(x_1)=f(x_2) \] Then: \[ \frac{x_1-2}{x_1-3}=\frac{x_2-2}{x_2-3} \] Cross multiply: \[ (x_1-2)(x_2-3)=(x_2-2)(x_1-3) \] Expand: \[ x_1x_2-3x_1-2x_2+6=x_1x_2-3x_2-2x_1+6 \] Simplify: \[ -x_1+x_2=0 \] \[ x_1=x_2 \]
Injection with Range \( \{a\} \) 📺 Video Explanation 📝 Question If: \[ f:A\to B \] is an injection and: \[ \text{Range}(f)=\{a\} \] determine the number of elements in set \(A\). ✅ Solution Given: \[ \text{Range}(f)=\{a\} \] This means every element of \(A\) maps to the same value \(a\). 🔹 Injection Property For injection: \[
Check Function \(f(x)=\dfrac{x}{x^2+1}\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=\frac{x}{x^2+1} \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Take: \[ x=2,\quad x=\frac12 \] Then: \[ f(2)=\frac{2}{5} \] \[ f\left(\frac12\right)=\frac{\frac12}{\frac14+1} =\frac{\frac12}{\frac54} =\frac{2}{5} \] Different inputs give same output. ❌ Not
Check Function \(f(x)=1+x^2\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=1+x^2 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Take: \[ x=2,\quad x=-2 \] Then: \[ f(2)=1+4=5,\quad f(-2)=1+4=5 \] But: \[ 2\neq-2 \] ❌ Not one-one. 🔹 Step 2: Check
Check Function \(f(x)=3-4x\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=3-4x \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ 3-4x_1=3-4x_2 \] So: \[ x_1=x_2 \] ✔ Function is one-one. 🔹 Step 2: Check Surjection
Check Function \(f(x)=5x^3+4\) on \( \mathbb{R} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{R}\to\mathbb{R},\quad f(x)=5x^3+4 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ 5x_1^3+4=5x_2^3+4 \] So: \[ x_1^3=x_2^3 \] Thus: \[ x_1=x_2 \] ✔ Function is one-one. 🔹
Check Function \(f(x)=x^3+1\) on \( \mathbb{Q} \) 📺 Video Explanation 📝 Question Check whether the function \[ f:\mathbb{Q}\to\mathbb{Q},\quad f(x)=x^3+1 \] is: injection (one-one) surjection (onto) bijection ✅ Solution 🔹 Step 1: Check Injection (One-One) Assume: \[ f(x_1)=f(x_2) \] Then: \[ x_1^3+1=x_2^3+1 \] So: \[ x_1^3=x_2^3 \] Thus: \[ x_1=x_2 \] ✔ Function is one-one. 🔹