inverse of f(x)=7x^-1/2+1

Question

find the inverse

Answer 1

f(x) = 7x^-(1/2) + 1

y = 7/sqr(x) + 1

to find the inverse, switch x and y

x = 7/sqr(y) + 1

solve for the new y

x-1 = 7/sqr(y)

sqr(y) = 7/(x-1)

square both sides

y = 49/(x-1)^2

y = 49/(x^2-2x+1)

f^-1(x) = 49/(x^2-2x+1) seems to be the inverse function

but by squaring both sides, it involved multiplying by a variable, which potentially introduces an extraneous solution.

f(x) is all in quadrant I

the inverse should also be all in quadrant I

but the above f^-1(x) is in both quadrant I and II

the actual inverse function should eliminate the quadrant II portion.

use a graphing calculator for f(x) and the above f^-1(x)

such as Desmos