Determine the domain and range of f(x)=-sqrt(x+3) and the equation of the inverse.
To determine the domain and range of the function f(x) = -√(x + 3), we need to consider the restrictions and possible output values.
1. Domain:
The domain represents the set of values for which the function is defined. In this case, the square root function is defined for non-negative values. Therefore, the expression x + 3 inside the square root must be greater than or equal to zero:
x + 3 ≥ 0
Solving this inequality, we find:
x ≥ -3
Hence, the domain of f(x) is all real numbers greater than or equal to -3.
2. Range:
The range represents the set of possible output values of the function. In this case, the square root function of any non-negative value is always non-negative or zero. Thus, the range of f(x) is all real numbers less than or equal to zero.
Domain: x ≥ -3
Range: f(x) ≤ 0
Now, let's find the equation of the inverse of f(x).
To find the inverse, interchange x and y and solve for y.
x = -√(y + 3)
To isolate y, square both sides of the equation:
x^2 = y + 3
Now, solve for y:
y = x^2 - 3
Hence, the equation of the inverse function is:
f^(-1)(x) = x^2 - 3