Question

Let f(x) = 4√(2x^2 - 1). Write an equation of the function obtained by reflecting f(x) in the x-axis and shifting it 56 units upward, do not simplify.

Answer 1

Supposed f(1) = 2, f(2) = 3, f(3) = 0, and g(1) = 7, g(2) = 1, g(3) = 4

Find:
g ○ f (2) =
f ○ f-1 (3) =
f ○ g-1 (4) =

Answer 2

reflection takes (x,y)->(x,-y)
translation takes (x,y)->(x,y+56)

So, g(x)
= -f(x)+56
= -4√(2x^2-1)+56

For the other set, note that
f^-1(3) = 2, since f(2) = 3
g^-1(4) = 3

So,

(g○f)(2) = g(f(2)) = g(3) = 4
By definition, (f○f^-1)(x) = x for any x
(f○g^-1)(4) = f(g^-1(4)) = f(3) = 0