If `f(x) = -5/7x-3` ` ` determine the slope of `f^(-1)(x)` without finding the equation `f^(-1)(x)` explicitly
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calendarEducator since 2012
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The inverse of a function is the reflection of the function in the line `y=x`.
The two gradients (slopes) multiply together to give the slope of the line
`y=x`, which equals 1.
Therefore `f' (f^(-1))' =1`
If the slope of `f` is `-5/7` then the slope of the inverse function`f^(-1)` is
`1/((-5/7)) = -7/5`
The slope of f^(-1) is one over the slope of f
So the slope of f^(-1) is 1/(-5/7) = -7/5
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calendarEducator since 2011
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starTop subjects are Math, Science, and Business
You should remember that the derivative of the function gives the slope of the tangent line at a point and the equation that relates the derivatives of the function and its inverse is `f'(x)*(f^(-1)(x))' = 1` .
Hence, you need to find dervative of the given function such that:
`f'(x)= (-5/7x - 3)' => f'(x) = -5/7*1 - 0 => f'(x) = -5/7`
Considering the equation `f'(x)*(f^(-1)(x))' = 1` yields:
`(f^(-1)(x))' = 1/f'(x) => (f^(-1)(x))' = 1/(-5/7)`
`(f^(-1)(x))' = -7/5`
Since the derivative of the function represents the slope, hence, evaluating the slope of the inverse function, whithout founding it, yields `(f^(-1)(x))' = -7/5.`
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