Find the domain and range of f superscript negative 1 baseline where lower f left parenthesis x right parenthesis equals Start Fraction 1 over 3 x plus 2 End Fraction.
To find the domain and range of the inverse function, we need to first find the inverse function of f(x) = 1/(3x+2).
To find the inverse function, we can interchange the x and y variables and solve for y.
x = 1/(3y+2)
Solving for y:
3y + 2 = 1/x
3y = 1/x - 2
y = (1/x - 2)/3
Therefore, the inverse function is f^(-1)(x) = (1/x - 2)/3.
Domain:
The inverse function is defined for all real numbers x except for the values that would make the denominator of the function equal to zero. In this case, the denominator is x, so the inverse function is defined for all real numbers x except for x = 0. Therefore, the domain of f^(-1)(x) is all real numbers except x = 0.
Domain: All real numbers except for x = 0.
Range:
To find the range, we need to analyze the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, (1/x - 2) approaches zero. As x approaches negative infinity, (1/x - 2) approaches negative infinity. Therefore, the range of f^(-1)(x) is all real numbers except for zero.
Range: All real numbers except for 0.
To find the domain and range of f^(-1)(x) = 1/(3x + 2), we first need to consider the domain and range of the function f(x) = 3x + 2.
Domain of f(x):
The domain of f(x) consists of all real numbers because there are no restrictions or limitations on the input value x.
Range of f(x):
The range of f(x) consists of all real numbers because for any real value of x, we can find a corresponding real value for f(x).
To find the domain and range of the inverse function f^(-1)(x), we swap the roles of the x and y variables. In other words, we interchange the domain and range of the original function f(x).
Domain of f^(-1)(x):
The domain of f^(-1)(x) is the range of the original function f(x), which is all real numbers.
Range of f^(-1)(x):
The range of f^(-1)(x) is the domain of the original function f(x), which is also all real numbers.
Therefore, the domain and range of f^(-1)(x) = 1/(3x + 2) are both all real numbers.