If f(x)= (-5/7)x - 3, determine the slope of `f^-1(x)` without finding the inverse.
The inverse of any relation (including functions) takes any point (x,y) in the relation and creates a point (y,x) in the inverse.
Thus if the slope of a function is given by `(Delta y)/(Delta x)` , then the slope of the inverse can be found by exchanging x and y; the slope of the inverse will be `(Delta x)/(Delta y)` which is the reciprocal of the slope of the original function.
The slope of `f(x)=(-5)/7 x +3` is `-5/7` , so the slope of the inverse is the reciprocal of `-5/7` which is `-7/5` .
To solve this problem first find two points that lie on the graph of the function f(x) = (-5/7)x - 3
Taking x = 0 , y = f(x) = (-5/7)x - 3 gives y = -3 and for x = 1, y = -26/7. The graph of f(x) passes through (0, -3) and (1, -26/7)
The graph of the inverse function `f^-1(x)` passes through the points (-3, 0) and (-26/7, 1)
This gives a slope of (1 - 0)/(-26/7 + 3) = -7/5
The slope of the graph representing `f^-1(x)` is `-7/5`