Check Injective / Surjective

🎥 Video Explanation

---

📝 Question

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

\[ f(x)=\frac{x^2-8}{x^2+2} \]

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

---

✅ Solution

🔹 Step 1: Check Injective

\[ f(x)=\frac{x^2-8}{x^2+2} \] depends on \(x^2\), so:

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

Different inputs give same output ⇒ ❌ Not one-one

—

🔹 Step 2: Find Range

Let:

\[ y=\frac{x^2-8}{x^2+2} \]

Solve for \(x^2\):

\[ y(x^2+2)=x^2-8 \]

\[ yx^2+2y=x^2-8 \]

\[ x^2(y-1)=-2y-8 \]

\[ x^2=\frac{-2y-8}{y-1} \] —

🔹 Step 3: Condition

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

\[ \frac{-2y-8}{y-1} \ge 0 \]

Solving gives:

\[ -4 \le y < 1 \]

—

🔹 Step 4: Check Onto

Range: \[ [-4,1) \]

Codomain is \(\mathbb{R}\), so not all real values covered.

❌ Not onto

—

🔹 Final Answer

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