Question

14 which of the following describes the graph of ( f^{-1}(x) )?

• the graph of ( f^{-1}(x) ) is the reflection of the graph of ( f(x) ) in the x - axis.
• the graph of ( f^{-1}(x) ) is the reflection of the graph of ( f(x) ) in the line ( y = -x ).
• the graph of ( f^{-1}(x) ) is the reflection of the graph of ( f(x) ) in the line ( y = x ).
• the graph of ( f^{-1}(x) ) is the reflection of the graph of ( f(x) ) in the y - axis.

Explanation:

Step1: Recall inverse function graph property

The key property of the graph of an inverse function \( f^{-1}(x) \) is that it is the reflection of the graph of \( f(x) \) over the line \( y = x \). This is because for a function \( y = f(x) \), its inverse \( x = f(y) \) (or \( y = f^{-1}(x) \)) swaps the \( x \)-and \( y \)-coordinates of each point on the graph of \( f(x) \), which geometrically corresponds to a reflection over \( y = x \).

Step2: Analyze each option

• Option 1: Reflection over \( x \)-axis changes \( (x,y) \) to \( (x, -y) \), not related to inverse functions. Eliminate.
• Option 2: Reflection over \( y = x \) swaps \( x \) and \( y \), matching the inverse function graph property.
• Option 3: Reflection over \( y=-x \) changes \( (x,y) \) to \( (-y, -x) \), not the inverse function reflection. Eliminate.
• Option 4: Reflection over \( y \)-axis changes \( (x,y) \) to \( (-x,y) \), not related to inverse functions. Eliminate.

Answer:

The graph of \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) in the line \( y = x \) (the second option from the left, depending on original option numbering; assuming the second option text is "The graph of \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) in the line \( y = x \)").