Determine the domain and range for the inverse of f(x)=1/4x+2
To determine the domain and range of the inverse of f(x) = 1/4x + 2, we first need to find the inverse function:
Let y = f(x)
1/4x + 2 = y
1/4x = y - 2
x = 4(y - 2)
x = 4y - 8
Now we switch x and y:
y = 4x - 8
The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.
For the original function f(x) = 1/4x + 2, the domain includes all real numbers except x = 0 (since we can't divide by zero). So the domain is (-∞, 0) U (0, ∞).
The range of the original function f(x) = 1/4x + 2 is all real numbers. Therefore, the range of the inverse function y = 4x - 8 is also all real numbers.
To find the domain and range of the inverse function, we first need to find the inverse of the given function, f(x) = 1/4x + 2.
Step 1: Replace f(x) with y.
y = 1/4x + 2
Step 2: Swap x and y.
x = 1/4y + 2
Step 3: Solve for y.
Multiply both sides by 4 to clear the fraction:
4x = y + 8
Subtract 8 from both sides:
4x - 8 = y
Rearrange to get y in terms of x, which is the inverse function:
y = 4x - 8
Now, we can determine the domain and range of the inverse function.
Domain:
The domain of the inverse function is the range of the original function. Since the original function is a linear function with a non-zero slope, the domain of the inverse function is all real numbers.
Range:
The range of the inverse function is the domain of the original function. Since the original function has a slope of 1/4, it is always increasing. Therefore, the range of the inverse function is also all real numbers.
In summary:
Domain of the inverse function: All real numbers
Range of the inverse function: All real numbers
To determine the domain and range of the inverse of a function, we can start by finding the inverse function and analyzing its properties.
Step 1: Find the inverse function of f(x).
To find the inverse function, we can swap the x and y variables and solve for y.
Let y = f(x) = 1/4x + 2.
Swap x and y:
x = 1/4y + 2
Next, solve for y:
x - 2 = 1/4y
4(x - 2) = y
y = 4x - 8
Thus, the inverse function of f(x) = 1/4x + 2 is f^(-1)(x) = 4x - 8.
Step 2: Determine the domain of f^(-1)(x).
The domain of a function refers to the set of all possible x-values for which the function is defined. In this case, the inverse function f^(-1)(x) = 4x - 8 does not have any restrictions on the x-values since it is defined for all real numbers. Therefore, the domain of the inverse function is (-∞, ∞).
Step 3: Determine the range of f^(-1)(x).
The range of a function represents all the possible y-values that the function can take. Since the inverse function is a line with a slope of 4, it will cover all possible y-values as x varies. In other words, the range of the inverse function is also (-∞, ∞).
To summarize:
- The domain of the inverse function f^(-1)(x) = 4x - 8 is (-∞, ∞).
- The range of the inverse function f^(-1)(x) = 4x - 8 is (-∞, ∞).