Answer: D
Explanation:
Because [tex]f^{-1}'(y)=\frac{1}{f'(x)}[/tex] and y=5 when x=10, [tex]f^{-1}'(5)=\frac{1}{f'(10)}=\frac{1}{8}[/tex]
The given equations are illustrations of inverse functions. The statement about the inverse of f that must be true is: [tex]f^{-1}(5) = \frac{1}{8}[/tex]
Given that:
[tex]f(10) = 5[/tex]
[tex]f'(10) = 8[/tex]
[tex]f(10) = 5[/tex] means that:
[tex]x = 10[/tex] and [tex]y = 5[/tex]
So, the relationship between the inverse of both functions is:
[tex]f^{-1}(y) = \frac{1}{f^{-1}(x)}[/tex]
Substitute values for x and y
[tex]f^{-1}(5) = \frac{1}{f^{-1}(10)}[/tex]
Substitute [tex]f'(10) = 8[/tex]
[tex]f^{-1}(5) = \frac{1}{8}[/tex]
Hence, option D is true, i.e.
[tex]D.\ f^{-1}(5) = \frac{1}{8}[/tex]
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