Find \(f^{-1}(x)\) for \(f(x)=10x-7\)
📝 Question
Let:
\[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=10x-7 \]
Find \(f^{-1}(x)\).
✅ Solution
🔹 Step 1: Check invertibility
The function \(f(x)=10x-7\) is linear with non-zero slope.
Hence, it is one-one and onto, so inverse exists.
—🔹 Step 2: Find inverse
Let:
\[ y=10x-7 \]
Interchange \(x\) and \(y\):
\[ x=10y-7 \]
Solve for \(y\):
::contentReference[oaicite:0]{index=0} —🔹 Step 3: Write inverse
\[ f^{-1}(x)=\frac{x+7}{10} \] —
🎯 Final Answer
\[ \boxed{f^{-1}(x)=\frac{x+7}{10}} \]
🚀 Exam Shortcut
- For \(ax+b\), inverse = \(\frac{x-b}{a}\)
- Swap \(x,y\) and solve
- Always divide by coefficient of \(x\)