(a) Find the inverse function of [tex]$f(x)=7 x-8$[/tex].
[tex]$f^{-1}(x)=\frac{x+8}{7}$[/tex]
(b) The graphs of [tex]$f$[/tex] and [tex]$f^{-1}$[/tex] are symmetric with respect to the line defined by [tex]$y=$[/tex] ______
Answer (2)
The problem provides a function f ( x ) and its inverse f − 1 ( x ) .
The key concept is that the graphs of a function and its inverse are symmetric with respect to the line y = x .
Therefore, the line of symmetry between the graphs of f ( x ) and f − 1 ( x ) is y = x .
The final answer is y = x .
Explanation
Understanding the Problem The problem asks us to find the line of symmetry between a function and its inverse, given that the function is f ( x ) = 7 x − 8 and its inverse is f − 1 ( x ) = 7 x + 8 . We need to identify the line with respect to which the graphs of f and f − 1 are symmetric.
Identifying the Line of Symmetry The graphs of a function and its inverse are always symmetric with respect to the line y = x . This is a fundamental property of inverse functions. To see why, consider a point ( a , b ) on the graph of f ( x ) . This means f ( a ) = b . For the inverse function, we have f − 1 ( b ) = a , which corresponds to the point ( b , a ) on the graph of f − 1 ( x ) . The points ( a , b ) and ( b , a ) are reflections of each other across the line y = x .
Conclusion Therefore, the graphs of f ( x ) and f − 1 ( x ) are symmetric with respect to the line y = x .
Examples
Imagine you are looking at a reflection in a mirror. The object and its reflection are symmetric with respect to the mirror's surface. Similarly, a function and its inverse are 'reflections' of each other across the line y = x . This concept is useful in various fields, such as physics (analyzing symmetrical phenomena), computer graphics (creating reflections and symmetrical designs), and cryptography (designing inverse operations for encoding and decoding messages).
To find the inverse of f ( x ) = 7 x − 8 , we get f − 1 ( x ) = 7 x + 8 . The graphs of f and f − 1 are symmetric with respect to the line y = x . Therefore, the answer is y = x .
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