Find \(f(x)\)

🎥 Video Explanation

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📝 Question

Given:

\[ g(x)=x^2+x-2 \]

\[ \frac{1}{2}g(f(x))=2x^2-5x+2 \]

Find \(f(x)\).

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✅ Solution

🔹 Step 1: Remove Fraction

\[ g(f(x))=4x^2-10x+4 \] —

🔹 Step 2: Substitute \(g(x)\)

\[ (f(x))^2 + f(x) -2 = 4x^2-10x+4 \] —

🔹 Step 3: Rearrange

\[ (f(x))^2 + f(x) – (4x^2-10x+6)=0 \] —

🔹 Step 4: Factor

Observe:

\[ 4x^2-10x+6=(2x-2)(2x-3) \]

So:

\[ (f(x))^2 + f(x) – (2x-2)(2x-3)=0 \]

Factor:

\[ (f(x)-(2x-2))(f(x)+(2x-3))=0 \] —

🔹 Step 5: Solve

\[ f(x)=2x-2 \quad \text{or} \quad f(x)=-2x+3 \] —

🔹 Final Answer

\[ \boxed{f(x)=2x-2 \text{ or } f(x)=3-2x} \]