Determing when f(x) and g(x) are in theta of h(x)

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In summary, the statement "If f(x) is in theta of h(x) and g(x) is in theta of h(x), then f(x)+g(x) is in theta of h(x)" is true. This is because the addition of two functions does not affect the value of the largest degree in both functions, as long as they are not multiplied. To formally prove this statement, one must start by defining what it means for a function to be in theta of another function.
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If f(x) is in theta of h(x) and g(x) is in theta of h(x), then is f(x)+g(x) in theta of h(x)?

My initial thoughts on this are yes since the addition of the two functions shouldn't impact the value of the largest degree in both functions, as they would if they were multiplied, but I'm wondering what sort of proof technique I could use to prove this in a more mathematically sound way.
 
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You are right about the answer. However, you can't talk about the largest degree because f(x) and g(x) are not necessarily polynomials. To prove the statement formally, start by writing what $f(x)\in\Theta(h(x))$ and $g(x)\in\Theta(h(x))$ means by definition.

I believe this answer is suitable for the Discrete Mathematics section of this forum because $\Theta$ is often used in measuring discrete resources consumed by algorithms.
 

FAQ: Determing when f(x) and g(x) are in theta of h(x)

What is the definition of theta in terms of functions?

Theta, denoted as Θ, is a mathematical notation used to represent the asymptotic behavior of functions. In other words, it describes the growth or rate of change of a function as its input value approaches infinity.

How do you determine when f(x) and g(x) are in theta of h(x)?

To determine when f(x) and g(x) are in theta of h(x), you must first find the limits of f(x) and g(x) as x approaches infinity. Then, if both limits are finite and equal, i.e. lim f(x) = lim g(x) = c, where c is a constant, then f(x) and g(x) are in theta of h(x).

What is the significance of determining theta of a function?

Determining theta of a function is important in understanding the complexity and efficiency of algorithms. It helps us analyze and compare different algorithms to see which one is more efficient in terms of time and space complexity.

Can two functions have different theta values?

Yes, two functions can have different theta values. Theta only represents the asymptotic behavior of a function, so two functions can have the same growth rate but different theta values. For example, f(x) = 2x and g(x) = x both have a growth rate of O(x) but have different theta values.

How does theta differ from big O and big Omega notation?

Theta notation differs from big O and big Omega notation in terms of their definitions and uses. While big O notation represents the upper bound of a function's growth rate, and big Omega notation represents the lower bound, theta notation represents the exact growth rate of a function. In other words, theta notation is more precise in describing the behavior of a function as its input value approaches infinity.

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