Check One-One / Onto

🎥 Video Explanation

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📝 Question

\[ f:\mathbb{R}\setminus\left\{\frac{3}{5}\right\} \to \mathbb{R}, \quad f(x)=\frac{3x+2}{5x-3} \]

Determine whether \(f\) is one-one and/or onto.

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✅ Solution

🔹 Step 1: Check Injective

Let \(f(x_1)=f(x_2)\):

\[ \frac{3x_1+2}{5x_1-3}=\frac{3x_2+2}{5x_2-3} \]

Cross multiply:

\[ (3x_1+2)(5x_2-3)=(3x_2+2)(5x_1-3) \]

Simplifying gives:

\[ x_1=x_2 \]

✔️ Function is one-one

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🔹 Step 2: Check Onto

Let:

\[ y=\frac{3x+2}{5x-3} \]

Solve for \(x\):

\[ y(5x-3)=3x+2 \]

\[ 5xy-3y=3x+2 \]

\[ x(5y-3)=3y+2 \]

\[ x=\frac{3y+2}{5y-3} \]

Defined only if \(5y-3\ne0\Rightarrow y\ne\frac{3}{5}\)

Range: \[ \mathbb{R}\setminus\left\{\frac{3}{5}\right\} \]

Codomain is \(\mathbb{R}\), so one value missing ⇒ ❌ Not onto

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🔹 Final Answer

\[ \boxed{\text{one-one but not onto}} \]