If f(x) and g(x) are inverse functions, in which line will g(x) be a reflection of f(x) ?

pan90

New member
Can anyone explain the logic behind the answer?

Taken from HiSet free practice test

MarkFL

Staff member
Suppose we have the point:

$$\displaystyle (x,f(x))$$

on the plot of $$f(x)$$. Then, on the plot of $$g(x)$$, we must have the corresponding point:

$$\displaystyle (f(x),x)$$

Now, consider that for all possible points, the locus of the mid-points is:

$$\displaystyle \left(\frac{x+f(x)}{2},\frac{x+f(x)}{2}\right)$$

Thereby implying that the line of symmetry must be:

$$\displaystyle y=x$$