Hello,
For a math assignment I have we're supposed to find out if the function f(x) = x^2- 2x + 3 has any of these properties:
-increasing/decreasing
-even/odd
-invertible or not
I said that it was increasing, even, and not invertible, but that was incorrect. Could someone please help?
You can easily tell that the graph is a parabola which opens up.
f(x) = (x-1)^2 + 2
So, it has a decreasing interval and an increasing interval.
It is decreasing on (-∞,1) and increasing on (1,∞)
You can do that without any calculus. But it's easy to verify since
f'(x) = 2x-2
f' < 0 for x < 1
f' > 0 for x > 1
It is neither even nor odd
not even, since f(-x) ≠ f(x) --- not all powers of x are even
not odd, since f(x) ≠ -f(x) -- not all powers of x are odd
Since it has two branches, it is not invertible unless you restrict the domain.
Also I asked a different Algebra question yesterday(17/6) at around 9 pm and it would be great if someone could offer some ideas on how to solve that algebraically.
Oh, I completely forgot that quadratics have both increasing and decreasing intervals! Thank you!
Of course, I'd be happy to help you with this math assignment!
To determine if the function f(x) = x^2 - 2x + 3 has the properties you mentioned, let's go step by step:
1. Increasing/Decreasing:
To determine if the function is increasing or decreasing, we need to look at the sign of its first derivative. If the first derivative is positive, the function is increasing. If the first derivative is negative, the function is decreasing.
To find the first derivative of f(x), we need to differentiate it with respect to x:
f'(x) = d/dx (x^2 - 2x + 3)
Differentiating each term, we get:
f'(x) = 2x - 2
Now, we can set f'(x) greater than zero or less than zero and solve for x:
2x - 2 > 0 or 2x - 2 < 0
By solving these inequalities, we can determine the intervals on which the function is increasing or decreasing.
2. Even/Odd:
A function is even if f(x) = f(-x) for every value of x. It means that the function is symmetric about the y-axis. On the other hand, if f(x) = -f(-x) for every value of x, the function is odd.
To check if f(x) is even or odd, we can substitute -x in place of x in the original function f(x) and see if it's equivalent.
3. Invertible or not:
A function is invertible if it has a well-defined inverse function. In other words, if for any output value, there is only one input value that corresponds to it.
To determine if f(x) is invertible, we need to check if it passes the horizontal line test. If there are any horizontal lines that intersect the graph of f(x) more than once, then it is not invertible.
By considering these three properties, we can analyze the function and determine if your original answers were correct or not.