find the inverse of the equation

Question

find the inverse of f(x)= √(8-4x)  -2 

 

find the domain and the range of the inverse function

Answer 1

y = f(x) = sqrt(8 - 4x) - 2

Domain: 8 - 4x >= 0, x <= 2.  Range: y >= -2.

 

To find inverse of f, change all x's to y's and all y's to x's:

x = sqrt(8 - 4y) - 2; Range: 8 - 4y >= 0, y <= 2; Domain: x >= -2

x + 2 = sqrt(8 - 4y)

(x + 2)^2 = 8 - 4y, x >= -2

y = f^-1(x) = 2 - (1/4)*(x + 2)^2, x >= -2.

Answer 2

y=√(8-4x)-2;

x=¼(8-(y+2)²)=1-y-y²/4;

 

Inverse of f(x) is:

 

1-x-x²/4;

Its range (-∞,2]. However, its domain must be restricted, otherwise this function is not one-to-one and cannot be the inverse of the original function. Formally its domain is R, but we restrict it in such a way so that it is equivalent to the range of original function.

 

Thus, its domain is [-2, ∞).