The function f(x)=(x-1)^2-4 is not one-to-one. If you restrict the domain for f(x) to x(less than or equal to symbol)1, what is its inverse function and the domain for the inverse?
To find the inverse function, we need to interchange the roles of x and f(x) and solve for x.
Let y = (x-1)^2 - 4.
Switch the roles:
x = (y-1)^2 - 4.
Now solve for y:
y = (x-1)^2 - 4.
This is the inverse function of f(x).
The domain for the inverse function is the range of f(x) with the domain restricted to x ≤ 1.
To find the range of f(x) with x ≤ 1, we need to find the minimum value of f(x) in that domain.
Taking the derivative of f(x):
f'(x) = 2(x-1).
Setting f'(x) = 0:
2(x-1) = 0,
x - 1 = 0,
x = 1.
So the minimum value occurs at x = 1.
Substituting x = 1 into f(x):
f(1) = (1-1)^2 - 4,
f(1) = -4.
Therefore, the range of f(x) with x ≤ 1 is (-∞, -4].
Therefore, the domain for the inverse function is (-∞, -4].
To find the inverse of a function, we need to interchange the variables x and y and solve the equation for y.
Let's start with the original function, f(x) = (x-1)^2 - 4.
Step 1: Interchange x and y: x = (y-1)^2 - 4.
Step 2: Solve for y: x + 4 = (y-1)^2.
Step 3: Take the square root of both sides: √(x + 4) = y - 1.
Step 4: Add 1 to both sides: y = √(x + 4) + 1.
This gives us the inverse function: f^(-1)(x) = √(x + 4) + 1.
Now, let's determine the domain for the inverse function. The domain of f(x) when x ≤ 1 is [-∞, 1] (all real numbers less than or equal to 1).
Therefore, the domain for the inverse function f^(-1)(x) is [-∞, 1].
So, the inverse function is f^(-1)(x) = √(x + 4) + 1, and its domain is [-∞, 1].
To find the inverse function, we need to switch the roles of x and y in the equation and solve for y. Let's start by switching the variables:
x = (y - 1)^2 - 4
Now, we can solve for y. Let's follow the steps:
1. Add 4 to both sides of the equation:
x + 4 = (y - 1)^2
2. Take the square root of both sides:
√(x + 4) = y - 1
3. Add 1 to both sides to isolate y:
√(x + 4) + 1 = y
Now, we have the inverse function:
f^(-1)(x) = √(x + 4) + 1
Next, let's determine the domain for the inverse function. Since we restricted the domain of f(x) to x ≤ 1, the range for f(x) is [-∞, -3]. Therefore, the domain for the inverse function will be the range of f(x). This means the domain will be [-∞, -3].
Hence, the inverse function of f(x) = (x - 1)^2 - 4, restricted to x ≤ 1, is given by f^(-1)(x) = √(x + 4) + 1, and its domain is [-∞, -3].