If `f(x) = -5/7x-3` ` `  determine the slope of `f^(-1)(x)` without finding the equation `f^(-1)(x)` explicitly

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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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.`