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find the inverse of the equation

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2 Answers

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, ∞).
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.