Remember that f(x) must be one-to-one (only one y-value for each x-value) over the domain where f –1(x)is defined as a function. So, in some cases you must restrict the domain of f(x) so that it's one-to-one. There might be more than one section of domain that's one-to-one.
HINT: The function f(x) is one-to-one wherever it's monotonic ("monotonic" simply means increasing or decreasing over the entire interval, so the derivative does not change sign).
To find these intervals, start by taking the first derivative, which is AP Calculus AB Semester 2 and setting it equal to zero.
I. (–∞, –3]
II. [–3, ∞)
III. [2, ∞)
I
II
III
Either I or II
Either I or III
Yes, it's Either 1 or 3, or answer E.
I believe that it is Either I or III
Actually, f(x) = x^3+3/2x^2-18x+4
When you differentiate to find critical points which are -3 & 2, you would see from (-infinity,-3] and [2,infinity). Those two ranges make f(x) become one-to-one function.
Well, well, well, it seems we've got some one-to-one fun going on here! Let's see which intervals can be called "monotonic" and satisfy the criteria.
For interval I, the function f(x) is decreasing, because it's from negative infinity (whoa, that's low!) up until -3 (which is also low, but in a different way). So, we can say that interval I is monotonic. One for I!
Now, for interval II, we've got the function f(x) increasing from -3 (which is like a turning point from low to higher grounds) all the way to positive infinity (that's the sky, baby!). This means interval II is monotonic too. Another one in the bag for II!
Lastly, we have interval III, where f(x) starts at 2 (a pretty modest beginning) and keeps on increasing throughout positive infinity (it's like Santa's reindeer flying higher and higher!). So, guess what? Interval III is monotonic too. Cha-ching for III!
To sum it up, all three intervals I, II, and III are monotonic and allow f(x) to be one-to-one. So, the answer is "Either I or II or III." Don't you just love having so many options? It's like a buffet of monotonic goodness!
Hope this explanation tickled your funny bone while also clarifying things. Take care and keep those monotonic vibes going!
To find the intervals where the function f(x) is one-to-one, we need to examine where the function is monotonically increasing or decreasing. We can do this by finding the intervals where the derivative of f(x) does not change sign.
To start, we take the first derivative of f(x) with respect to x and set it equal to zero to find critical points.
Let's assume that the first derivative is denoted by f'(x). So we have f'(x) = 0.
Once we find the critical points, we can determine the intervals where f(x) is monotonic by checking the signs of the first derivative in each interval.
In this case, the given options are:
I. (–∞, –3]
II. [–3, ∞)
III. [2, ∞)
To determine if any of these options are correct, we need to find the critical points by solving f'(x) = 0. Once we have the critical points, we can check the signs of f'(x) in each interval to see if the function is increasing or decreasing.
Once we find the intervals where the function is increasing or decreasing, we can determine which options, if any, are correct.