Question

Consider the tables that represent a continuous function and its inverse.



Which is an accurate comparison of the functions?

The reciprocal of the slope of f(x) is the same as the slope of f–1(x).
The negative reciprocal of the slope of f(x) is the same as the slope of f–1(x).
The y-coordinate of the y-intercept of f(x) is the same as the y-coordinate of the y-intercept of f–1(x).
The opposite of the y-coordinate of the y-intercept of f(x) is the same as the x-coordinate of the x-intercept of f–1(x).

Answer 1

Answer: The reciprocal of the slope of f(x) is the same as the slope of f⁻¹(x).

Reason: If f contains two points (x1,y1) and (x2,y2), then f⁻¹ contains (y1,x1) and (y2,x2). Slope of f = (y2−y1)/(x2−x1). Slope of f⁻¹ = (x2−x1)/(y2−y1), which is the reciprocal of the slope of f (provided neither slope is 0).

Notes: "Negative reciprocal" is the condition for perpendicular lines, not inverses. Also the y-intercept of f (0,b) becomes the x-intercept b of f⁻¹, so y-intercepts are not generally equal.