Check Injective / Surjective
🎥 Video Explanation
📝 Question
Let \( f:\mathbb{R} \to \mathbb{R} \),
\[ f(x)=6^x+6^{|x|} \]
- A. one-one and onto
- B. many-one and onto
- C. one-one and into
- D. many-one and into
✅ Solution
🔹 Step 1: Case-wise Form
Case 1: \(x \ge 0\)
\[ f(x)=6^x + 6^x = 2\cdot 6^x \]
Case 2: \(x < 0\)
\[ f(x)=6^x + 6^{-x} \]
—🔹 Step 2: Check Injective
For \(x \ge 0\): increasing
For \(x < 0\): symmetric-type behavior
Example:
\[ f(1)=2\cdot6=12 \]
\[ f(-1)=\frac{1}{6}+6=\frac{37}{6} \]
Near zero:
\[ f(0)=2 \]
Function is not strictly monotonic ⇒ ❌ Not one-one
—🔹 Step 3: Range
Minimum at \(x=0\):
\[ f(0)=2 \]
As \(x \to \infty\), \(f(x)\to \infty\)
Range: \[ [2,\infty) \]
—🔹 Step 4: Check Onto
Codomain is \(\mathbb{R}\), but values < 2 not covered.
❌ Not onto
—🔹 Final Answer
\[ \boxed{\text{Option D: many-one and into}} \]