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) = -sqrt(x+3), we need to consider the restrictions on the input (x) and the resulting output (f(x)).
1. Domain:
The domain refers to all the possible values of x for which the function is defined. In this case, the square root function has a restriction that the expression inside the square root (x+3) must be greater than or equal to zero. So, to find the domain, we set the expression x+3 greater than or equal to zero and solve for x:
x + 3 ≥ 0
x ≥ -3
Therefore, the domain of f(x) = -sqrt(x+3) is all real numbers greater than or equal to -3, or written in interval notation: (-3, ∞).
2. Range:
The range refers to all the possible values of f(x) or the output of the function. Since the square root function always returns a non-negative value, the range of f(x) is all real numbers greater than or equal to zero, or written in interval notation: [0, ∞).
Now, let's find the equation of the inverse of f(x).
To find the inverse of a function, we need to interchange the x and y variables and solve for y.
Let y = f(x) = -sqrt(x+3)
Interchanging x and y gives us:
x = -sqrt(y+3)
To solve this equation for y, we need to isolate the square root.
x = -sqrt(y+3)
x² = (sqrt(y+3))²
x² = y + 3
y = x² - 3
Therefore, the equation of the inverse function of f(x) = -sqrt(x+3) is f^(-1)(x) = x^2 - 3.