Check Injective / Surjective

🎥 Video Explanation

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📝 Question

Let \( f:\mathbb{R} \to \mathbb{R} \) be defined by

\[ f(x)=\frac{e^{x^2}-e^{-x^2}}{e^{x^2}+e^{-x^2}} \]

• A. one-one but not onto
• B. one-one and onto
• C. onto but not one-one
• D. neither one-one nor onto

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✅ Solution

🔹 Step 1: Simplify

\[ f(x)=\tanh(x^2) \]

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🔹 Step 2: Check Injective

\[ f(x)=f(-x) \]

Different inputs → same output ⇒ ❌ Not one-one

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🔹 Step 3: Range

Since \(x^2 \ge 0\):

\[ f(x)=\tanh(x^2) \]

\[ \tanh(t), \; t \ge 0 \Rightarrow [0,1) \]

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🔹 Step 4: Check Onto

Range: \[ [0,1) \]

Codomain is \(\mathbb{R}\)

❌ Not onto

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🔹 Final Answer

\[ \boxed{\text{Option D: neither one-one nor onto}} \]