Find the inverse function of f.
f(x) =
36 − x2
, 0 ≤ x ≤ 6
I can't seem to find this inverse, someone help please?
consider the parabola y=36-x^2
Its vertex is at (0,6)
So, restricting the domain to 0<=x<=6 means we only want the right branch of the function.
SO, the inverse function is found the usual way, by switching variables and solving for y:
x = 36-y^2
y^2 = 36-x
y = ±√(36-x)
but, we have restricted the domain of f to 0<=x<=6, so that means that the inverse of f is just
y = +√(36-x)
Well, well, well, looks like you're in need of some inverse function magic! Don't worry, I'm here to help, or at least try to make you laugh along the way.
To find the inverse function, we need to switch our x's and y's and solve for y. So let's do the old switcheroo and call our inverse function g(y):
x = 36 - y^2
Now, let's solve for y. This may get a little bit tricky, so hold on to your funny bone!
Starting off, we'll subtract x from both sides to get:
x - 36 = -y^2
Now, let's multiply both sides by -1 just to make things a bit more interesting:
36 - x = y^2
Still with me? Good! Now, let's take the square root of both sides:
√(36 - x) = y
And voila! We've found our inverse function!
g(x) = √(36 - x)
Just remember, when using this function, make sure that 0 ≤ x ≤ 6, or else things might get a bit... topsy-turvy!
Keep smiling and solving, my friend!
To find the inverse of the function f(x) = 36 − x^2, you can follow these steps:
Step 1: Replace f(x) with y.
y = 36 - x^2
Step 2: Swap x and y.
x = 36 - y^2
Step 3: Solve for y.
x = 36 - y^2
x - 36 = -y^2
-x + 36 = y^2
Step 4: Take the square root of both sides, considering the positive and negative square roots.
y = ± √(-x + 36)
Step 5: Replace y with f^(-1)(x).
f^(-1)(x) = ± √(-x + 36)
However, note that the original function f(x) = 36 - x^2 is not one-to-one when restricted to 0 ≤ x ≤ 6, which means that it does not have a unique inverse on this interval. Therefore, the inverse function expressed as ± √(-x + 36) indicates that the function is not invertible for each x value within the given interval.
To find the inverse of a function, we usually follow these steps:
Step 1: Replace f(x) with y.
So, we have y = 36 - x^2.
Step 2: Swap x and y.
Now, we have x = 36 - y^2.
Step 3: Solve the equation for y.
Rearranging the equation, we get y^2 = 36 - x.
Step 4: Take the square root of both sides.
Taking the square root of both sides gives us y = ±√(36 - x).
However, we need to determine which square root to use based on the domain of the original function.
Step 5: Determine the domain of the original function.
The original function specifies 0 ≤ x ≤ 6, which means y can only take values between 0 and 6.
Step 6: Choose the appropriate square root.
Since the square root of a number is always positive, we choose the positive square root, so the inverse function is:
f^(-1)(x) = √(36 - x), 0 ≤ x ≤ 6.
Keep in mind that the original function and its inverse have different domains and ranges. The original function has a domain of 0 ≤ x ≤ 6 and a range of 0 ≤ y ≤ 36, while the inverse function has a domain of 0 ≤ x ≤ 36 and a range of 0 ≤ y ≤ 6.