Let f(x)=11/x+7 with the domain [-2,3]. Find the domain of f^-1.
I bet you mean f(x) = 11/(x+7) with domain (-2 , 3)
find inverse function
x = 11/(y+7)
x y + 7 x = 11
xy = 11 - 7x
y = (11 - 7x)/x
when y = -2
-2 x = 11 - 7 x
x = 11/5
when y = 3
3 x = 11 - 21
x = -10/3
so domain would be from -10/3 to 11/5 but the inverse is undefined at x = 0
In the original:
if x= -2 , y = 11/5
if x = 3, y = 11/10
So the curve is continuous from (-2,11/5) to (3,11/10)
in the inverse the domain of the original becomes the range of the inverse, and the range of the original becomes the domain of the inverse
domain of inverse : [11/10 , 11/5]
make a rough sketch, remember that that inverse of a function results in a reflection in the line y = x
To find the domain of the inverse function, f^(-1), we need to determine the range of the original function, f(x).
Given that f(x) = 11/x + 7 with the domain [-2,3], we need to consider any restrictions on x that may cause the function to be undefined.
Since the only restriction is in the denominator, set the denominator equal to zero and solve for x:
x + 7 = 0
x = -7
Therefore, x = -7 is the only value that makes the function undefined. Thus, the range of f(x) is all real numbers except for -7.
Since the domain of f^(-1) is the range of f(x), the domain of f^(-1) is all real numbers except for -7.
Therefore, the domain of f^(-1) is (-∞, -7) U (-7, ∞).