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Chapter 5: Problem 84

Describe the relationship between the graph of a function and the graph of its inverse function.

Short Answer

Expert verified
The graph of an inverse function is a reflection of the graph of the original function over the line y = x. Hence, any point present on the graph of the original function will have its coordinates interchanged in the graph of the inverse function.

Step by step solution

01

Defining a Function and its Inverse

Conceptually, a function is basically a rule that assigns each input to a unique output. An inverse function, on the other hand, is a rule that operates in the reverse manner, it takes outputs of the original function as inputs and gives corresponding inputs of the original function as outputs. It is denoted by \( f^{-1}(x) \). Thus, a function and its inverse undo each other.
02

Graphical Representation of Function and Inverse Function

Graphically, considering any function \( f(x) \), its graph will be a set of points in the Cartesian coordinate system such that for each \( x \), there is a related \( y \) value. For the inverse function \( f^{-1}(x) \), it will take the \( y \) values of the original function and associate with them their corresponding \( x \) values.
03

The Relationship between Function Graph and its Inverse

The graph of an inverse function is a reflection of the original function's graph over the line \( y = x \). The x-coordinates and y-coordinates of each point on the original function's graph are interchanged in the inverse function's graph. Hence, if \( (a, b) \) is a point on the graph of \( f(x) \), then \( (b, a) \) will be a point on the graph of \( f^{-1}(x) \).

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