Find \(f^{-1}(x)\) and Range of \(f(x)=\frac{2x}{5x+3}\)
📝 Question
Let:
\[ f:\mathbb{R}\setminus\left\{-\frac{3}{5}\right\}\to \mathbb{R}, \quad f(x)=\frac{2x}{5x+3} \]
Find \(f^{-1}(x)\) and the range of \(f\).
✅ Solution
🔹 Step 1: Find inverse
Let:
\[ y=\frac{2x}{5x+3} \]
Cross multiply:
\[ y(5x+3)=2x \]
\[ 5xy+3y=2x \]
\[ 3y=x(2-5y) \]
\[ x=\frac{3y}{2-5y} \]
Interchange \(x\) and \(y\):
:contentReference[oaicite:0]{index=0} —🔹 Step 2: Find range
From inverse expression, denominator must not be zero:
\[ 2-5x \ne 0 \]
\[ x \ne \frac{2}{5} \]
Hence, range of \(f\) is:
\[ \mathbb{R}\setminus\left\{\frac{2}{5}\right\} \] —
🎯 Final Answer
\[ \boxed{f^{-1}(x)=\frac{3x}{2-5x}} \]
\[ \boxed{\text{Range}=\mathbb{R}\setminus\left\{\frac{2}{5}\right\}} \]
🚀 Exam Shortcut
- Cross multiply and solve for \(x\)
- Swap variables to get inverse
- Denominator ≠ 0 ⇒ excluded value in range