Determine if each of the following functions is in Big-O, is Big-Omega and is Big-Theta of x: (Could be multiple)
a) f(x)=10
b) f(x)=3x+7
C) f(x)=x^2+x+1
d) f(x)=5log(x)
e) f(x)=|x|
f) f(x)=x/2
I'm in calc 1 and I have no idea what's it even asking me. Help? Thanks in advance!
No problem! Let's break down the notation and concepts involved.
When we say a function f(x) is in Big-O notation, it essentially means that f(x) has an upper bound. More formally, if f(x) is Big-O of g(x), it means that there exists a positive constant c, such that f(x) is always less than or equal to c * g(x) for sufficiently large x.
On the other hand, if f(x) is in Big-Omega notation, it means that f(x) has a lower bound. To be more precise, if f(x) is Big-Omega of g(x), there exists a positive constant c, such that f(x) is always greater than or equal to c * g(x) for sufficiently large x.
Lastly, when we say f(x) is in Big-Theta notation, it means that f(x) has both an upper and a lower bound. Formally, f(x) is Big-Theta of g(x) if there exist positive constants c1 and c2, such that f(x) is always bounded between c1 * g(x) and c2 * g(x) for sufficiently large x.
Now let's determine the Big-O, Big-Omega, and Big-Theta relationships for each function you've listed:
a) f(x) = 10
This function is a constant, so it is always the same no matter the value of x. Therefore, f(x) = 10 is both in Big-O, Big-Omega, and Big-Theta of x^0, which is another constant.
b) f(x) = 3x + 7
This function is a linear function, meaning its growth is directly proportional to x. As x gets larger, the coefficient of x (3 in this case) becomes less significant. Therefore, f(x) = 3x + 7 is in Big-O, Big-Omega, and Big-Theta of x.
c) f(x) = x^2 + x + 1
This function is a quadratic function, meaning its growth is proportional to the square of x. Compared to the linear term (x), the quadratic term (x^2) dominates as x becomes larger. Therefore, f(x) = x^2 + x + 1 is in Big-O, Big-Omega, and Big-Theta of x^2.
d) f(x) = 5log(x)
This function is a logarithmic function, which grows slowly as x increases. Logarithm functions are typically bounded by linear functions for sufficiently large x. Therefore, f(x) = 5log(x) is in Big-O of x, but not in Big-Omega or Big-Theta of x.
e) f(x) = |x|
This function represents the absolute value of x. It grows linearly as x becomes larger, regardless of whether x is positive or negative. Therefore, f(x) = |x| is in Big-O, Big-Omega, and Big-Theta of x.
f) f(x) = x/2
This function is a linear function with a coefficient of 1/2. As x gets larger, the coefficient, in this case, becomes less significant. Therefore, f(x) = x/2 is in Big-O, Big-Omega, and Big-Theta of x.
To summarize:
a) f(x) = 10: Big-O, Big-Omega, and Big-Theta of x^0.
b) f(x) = 3x + 7: Big-O, Big-Omega, and Big-Theta of x.
c) f(x) = x^2 + x + 1: Big-O, Big-Omega, and Big-Theta of x^2.
d) f(x) = 5log(x): Big-O of x, not in Big-Omega or Big-Theta of x.
e) f(x) = |x|: Big-O, Big-Omega, and Big-Theta of x.
f) f(x) = x/2: Big-O, Big-Omega, and Big-Theta of x.
I hope this explanation helps you understand the concepts of Big-O, Big-Omega, and Big-Theta notations!
If you truly have no idea, you clearly have not studied your text. In this case, google is your friend. It will provide many discussions (wikipedia is a good one), as well as youtube videos.
When you are really lost, and need general instruction, start with google. This site is mainly for answering specific questions.