Question

which statement verifies that \\(f(x)\\) and \\(g(x)\\) are inverses of each other?

\\(f(g(x)) = x\\)

\\(f(g(x)) = x\\) and \\(g(f(x)) = -x\\)

\\(f(g(x)) = \frac{1}{g(f(x))}\\)

\\(f(g(x)) = x\\) and \\(g(f(x)) = x\\)

Explanation:

Define inverse function verification

$$ f(g(x)) = x \quad \text{and} \quad g(f(x)) = x $$

Evaluate the given options

• Option 1: \(f(g(x)) = x\) is incomplete because verification requires composition in both directions.
• Option 2: \(f(g(x)) = x\) and \(g(f(x)) = -x\) is incorrect because the second composition must equal \(x\), not \(-x\).
• Option 3: \(f(g(x)) = \frac{1}{g(f(x))}\) is incorrect as it represents a reciprocal relationship, not an inverse function relationship.
• Option 4: \(f(g(x)) = x\) and \(g(f(x)) = x\) correctly states the algebraic definition of inverse functions.

Answer:

• (A) \(f(g(x)) = x\)
• (B) \(f(g(x)) = x\) and \(g(f(x)) = -x\)
• (C) \(f(g(x)) = \frac{1}{g(f(x))}\)
• (D) \(f(g(x)) = x\) and \(g(f(x)) = x\) (Correct answer)