CAT 2017

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Composition of Functions
• Formula: $$f(x) = x^{2}, \quad g(x) = 2^{x}, \quad \text{Evaluate } f\big[f(g(x)) + g(f(x))\big] \text{ at } x = 1$$

2. Calculation

We are given (f(x) = x^2) and (g(x) = 2^x). We need to evaluate (f\big[f(g(x)) + g(f(x))\big]) at (x = 1).

Step 1: Compute (g(x)) at (x = 1): $$g(1) = 2^{1} = 2$$

Step 2: Compute (f(g(1))): $$f(g(1)) = f(2) = 2^{2} = 4$$

Step 3: Compute (f(x)) at (x = 1): $$f(1) = 1^{2} = 1$$

Step 4: Compute (g(f(1))): $$g(f(1)) = g(1) = 2^{1} = 2$$

Step 5: Add the two results: $$f(g(1)) + g(f(1)) = 4 + 2 = 6$$

Step 6: Apply (f) to the sum: $$f\big[f(g(1)) + g(f(1))\big] = f(6) = 6^{2} = 36$$

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3. Solution

Answer = Option C ✅

The final calculated value is 36.