Question

If \(f(x) = x^2\) and \(g(x) = [x^2]\) where \([\cdot ]\) represents the greatest integer function then, \((f\circ g)(\frac{3}{2})+(g\circ f)(\frac{3}{2})\) is equal to ...

• A. \(2\)
• B. \(5\)
• C. \(9\)
• D. \(10\)

Hint:

Compute each composite carefully, applying the inner function first.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
\(f\circ g\) means \(f(g(x))\) and \(g\circ f\) means \(g(f(x))\). The inner function is applied first.

Step 2: Compute (f o g)(3/2):
\(g(3/2)=[(3/2)^2]=[2.25]=2\). Then \(f(2)=2^2=4\).

Step 3: Compute (g o f)(3/2):
\(f(3/2)=\dfrac94=2.25\). Then \(g(2.25)=[(2.25)^2]=[5.0625]=5\).

Step 4: Add:
\[ 4+5=9 \]
Option (A) 2 and (D) 10 come from using the wrong order of composition or from dropping the greatest integer. The correct sum is 9.

Final Answer:
The sum is 9, option (C). \[ \boxed{\text{(C) } 9} \]