Check One-One / Onto
🎥 Video Explanation
📝 Question
Given \( f:\mathbb{R} \to \mathbb{R} \),
\[ f(x)=2^x + 2^{|x|} \]
Find the correct option:
- 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 Expression
Case 1: \(x \ge 0\)
\[ f(x)=2^x + 2^x = 2^{x+1} \]
Case 2: \(x < 0\)
\[ f(x)=2^x + 2^{-x} \] —
🔹 Step 2: Check One-One
For \(x \ge 0\): strictly increasing.
For \(x < 0\): symmetric behavior exists.
Example:
\[ f(1)=2^2=4 \]
\[ f(-1)=2^{-1}+2^1=\frac{1}{2}+2=\frac{5}{2} \]
But near zero:
\[ f(0)=2 \]
Function decreases on negative side and increases on positive side ⇒ repeats values.
❌ Not one-one ⇒ many-one
—🔹 Step 3: Check Onto
Minimum value occurs at \(x=0\):
\[ f(0)=2 \]
So range: \[ [2, \infty) \]
Codomain is \(\mathbb{R}\), but negative and values less than 2 are not covered.
❌ Not onto
—🔹 Final Answer
\[ \boxed{\text{Option D: many-one and into}} \]