Consider the function f(x)=4 - x^2 for the domain [ 0,∞).
Find f^−1 (x), where f^−1 is the inverse of f.
Also state the domain of f^−1 in interval notation.
f^-1 (x) = ___ for the domain ___
swap variables
x = 4-y^2
y = ±?(4-x)
But you need to pick the branch where 4-x >= 0.
take a look at the graph and I'm sure you can answer the questions.
http://www.wolframalpha.com/input/?i=%E2%88%9A(4-x)
To find the inverse of the function f(x) = 4 - x^2, we will follow these steps:
Step 1: Replace f(x) with y to get the equation in terms of y: y = 4 - x^2.
Step 2: Swap x and y to get the equation in terms of x: x = 4 - y^2.
Step 3: Solve for y by rearranging the equation: y^2 = 4 - x.
Step 4: Take the square root of both sides to isolate y: y = ±√(4 - x).
Since f(x) = 4 - x^2 represents a parabola opening downwards, the inverse relation will be a reflection of the original function over the line y = x. Therefore, we will only consider the positive square root: y = √(4 - x).
Thus, the inverse function is f^−1(x) = √(4 - x).
To find the domain of the inverse function, we need to examine the restrictions on x. In the original function, the domain was [0, ∞). However, when finding the inverse, we need to consider the range of the original function as the domain of the inverse.
In f(x) = 4 - x^2, the range is [-∞, 4]. Therefore, the domain of the inverse function, f^−1(x), is [-∞, 4].
To find the inverse of a function, we need to swap the roles of x and y and solve for y. Let's start by swapping x and y in the equation f(x) = 4 - x^2:
x = 4 - y^2
Now, let's solve for y:
y^2 = 4 - x
Taking the square root of both sides:
y = ±√(4 - x)
Since f(x) = 4 - x^2 has a restriction on the domain [0, ∞), we need to consider the corresponding range for the inverse function f^−1(x). For the inverse of f(x), the new domain will be the range of f(x) [0, ∞). Thus, the domain of f^−1 is [0, ∞) in interval notation.
So, the inverse function f^−1(x) is:
f^−1(x) = ±√(4 - x)
with a domain of [0, ∞).