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.
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) =
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
To reflect a function in the x-axis, we need to change the sign of the y-values. To shift a function upward, we add a constant value to the y-values.
Let's start by reflecting the function f(x) = 4√(2x^2 - 1) in the x-axis. We change the sign of the y-values by multiplying by -1.
g(x) = -4√(2x^2 - 1)
Now, to shift g(x) 56 units upward, we add 56 to the y-values.
h(x) = -4√(2x^2 - 1) + 56
Therefore, the equation of the function obtained by reflecting f(x) in the x-axis and shifting it 56 units upward is h(x) = -4√(2x^2 - 1) + 56.