let f(x)=x+2/x+8. what does f^-1(-4) equal?
To find the value of f^-1(-4), we need to find the inverse function of f(x), where f^-1(x) essentially undoes what f(x) does.
First, let's find the inverse function of f(x).
Step 1: Replace f(x) with y.
y = (x + 2)/(x + 8)
Step 2: Swap x and y.
x = (y + 2)/(y + 8)
Step 3: Solve for y.
Multiply both sides by (y + 8):
x(y + 8) = (y + 2)
Distribute:
xy + 8x = y + 2
Move all terms containing y to one side:
xy - y = 2 - 8x
Factor out y:
y(x - 1) = 2 - 8x
Divide both sides by (x - 1):
y = (2 - 8x)/(x - 1)
Now we know the inverse function of f(x) is f^-1(x) = (2 - 8x)/(x - 1).
To find the value of f^-1(-4), substitute -4 into the inverse function equation.
f^-1(-4) = (2 - 8(-4))/(-4 - 1)
= (2 + 32)/(-5)
= 34/(-5)
= -6.8
Therefore, f^-1(-4) is equal to -6.8.