Question

f : R → R and g : [0, ∞) → R are defined by f(x) = x² and g(x)=\(\sqrt{x}\). Which one of the following is not true ?

• A. (fog)(2) = 2
• B. (gof)(4) = 4
• C. (gof)(-2) = 2
• D. (fog)(-4) = 4

Hint:

When working with composite functions, always check the domain of each function. In this case, \( g(x) = \sqrt{x} \) is only defined for \( x \geq 0 \), so evaluating \( g(-4) \) is not valid. Ensure that the input to the composite function falls within the domain of the inner function.

Answer 1

The Correct Option is D

The correct answer is (D) : (fog)(-4) = 4.

Answer 2

The correct answer is: (D) (fog)(-4) = 4.

We are given two functions:

• f(x) = x² (a function from \( \mathbb{R} \) to \( \mathbb{R} \))
• g(x) = \( \sqrt{x} \) (a function from \([0, \infty) \) to \( \mathbb{R} \))

We need to evaluate the composite function \( (f \circ g)(x) \), which means applying \( g(x) \) first, and then applying \( f(x) \) to the result of \( g(x) \). The composite function is:

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

Now, let’s evaluate \( (f \circ g)(-4) \): - First, we apply \( g(x) \) to \( -4 \): Since \( g(x) = \sqrt{x} \), \( g(-4) \) is undefined because the square root of a negative number is not real. - Therefore, \( (f \circ g)(-4) \) is undefined, not 4. Thus, the statement \( (f \circ g)(-4) = 4 \) is not true. Therefore, the correct answer is (D) (fog)(-4) = 4.