The function is not one-to-one. If you restrict the domain for to , what is its inverse function and the domain for the inverse? (1 point) Responses Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4 Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 plus Start Root x plus 4 End Root, x greater-than-or-equal-to negative 4 Image with alt text: lower f superscript negative 1 baseline left-parenthesis x right-parenthesis equals 1 minus Start Root x minus 4 End Root, x greater-than-or-equal-to 4
The correct inverse function is:
f^(-1)(x) = 1 - √(x + 4), x ≥ -4
The inverse function of the given function is f^(-1)(x) = 1 - √(x + 4), where x ≥ -4.
To find the inverse function of a given function, you need to follow these steps:
1. Replace the function notation "f(x)" with "y".
2. Swap the variables "x" and "y". In other words, replace "y" with "x" and "x" with "y".
3. Solve the resulting equation for "y" to obtain the inverse function.
4. Replace "y" with the inverse function notation "f^(-1)(x)" to represent the inverse function.
In this case, the given function is not one-to-one, meaning it fails the horizontal line test. However, you can still find its restricted inverse function and its domain by following the steps mentioned above.
1. Replace "f(x)" with "y":
y = 1 - √(x + 4) (Option 1)
y = 1 + √(x + 4) (Option 2)
y = 1 - √(x - 4) (Option 3)
2. Swap the variables "x" and "y":
x = 1 - √(y + 4) (Option 1)
x = 1 + √(y + 4) (Option 2)
x = 1 - √(y - 4) (Option 3)
3. Solve for "y" to obtain the inverse function:
For Option 1:
x = 1 - √(y + 4)
√(y + 4) = 1 - x
y + 4 = (1 - x)^2
y = (1 - x)^2 - 4
The inverse function (Option 1) is f^(-1)(x) = (1 - x)^2 - 4
For Option 2:
x = 1 + √(y + 4)
√(y + 4) = x - 1
y + 4 = (x - 1)^2
y = (x - 1)^2 - 4
The inverse function (Option 2) is f^(-1)(x) = (x - 1)^2 - 4
For Option 3:
x = 1 - √(y - 4) (This equation doesn't correspond to the given options, so it is incorrect.)
4. Determine the domain for the inverse function:
The domain for the inverse function is determined by the range of the original function. Since the original function is not one-to-one, it does not have an inverse on its entire domain. However, when restricting the domain, the inverse function exists for x greater than or equal to -4.
Therefore, the correct answer is:
The inverse function is f^(-1)(x) = (1 - x)^2 - 4, and its domain is x greater than or equal to -4.