Question

Let f(x) = x² and \(g(x) = \sqrt{9+x}\). Then the value of \((f^\circ g-g^\circ f)(4)\) is equal to

• A. 6
• B. √6
• C. √8
• D. 8
• E. 5

Answer 1

The Correct Option is D

We are given the functions:

• \( f(x) = x^2 \) 
• \( g(x) = \sqrt{9 + x} \)

We need to find the value of \( (f \circ g - g \circ f)(4) \). This represents the difference between the composition of \( f \) and \( g \) and the composition of \( g \) and \( f \) evaluated at \( x = 4 \).

First, let's compute \( f \circ g \) and \( g \circ f \):

• \( (f \circ g)(x) = f(g(x)) = f(\sqrt{9 + x}) = (\sqrt{9 + x})^2 = 9 + x \)
• \( (g \circ f)(x) = g(f(x)) = g(x^2) = \sqrt{9 + x^2} \)

Now, we compute the value of \( (f \circ g - g \circ f)(4) \):

• \( (f \circ g)(4) = 9 + 4 = 13 \)
• \( (g \circ f)(4) = \sqrt{9 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Therefore, \( (f \circ g - g \circ f)(4) = 13 - 5 = 8 \).

The correct answer is 8.