Question

if \\(f(x)\\) and \\(g(x)\\) are inverse functions of each other, which of the following shows the graph of \\(f(g(x))\\)?

Explanation:

Define the composite function of inverses

Using the Inverse Functions and Composite Functions knowledge points:
If \(f(x)\) and \(g(x)\) are inverse functions of each other, then by definition, their composition yields the identity function:

$$ f(g(x)) = x $$

for all \(x\) in the domain of \(g(x)\).

Identify the graph of the identity function

The equation \(y = f(g(x))\) simplifies directly to:

$$ y = x $$

This represents a straight line passing through the origin \((0,0)\) with a slope of \(1\).

Analyze the given options

Let's examine the visible graphs:

• The first option shows a line passing through \((0,0)\) with a slope of \(0.5\) (it passes through \((2,1)\) and \((4,2)\)), which represents \(y = 0.5x\).
• The second option (partially visible at the bottom) shows a line with a slope of \(1\) passing through the origin, representing \(y = x\).

Therefore, the correct graph is the line \(y = x\).

Answer:

• (A) A line passing through the origin with a slope of 0.5, representing \(y = 0.5x\)
• (B) A line passing through the origin with a slope of 1, representing \(y = x\) (Correct answer)