Answered

To estimate [tex]O[/tex] of a given function, pick the highest order term and ignore the coefficient.

Example:
[tex]\[
\begin{array}{l}
T(n)=3n+5 \rightarrow O(n) \\
T(n)=3n^2+4n+2 \rightarrow O\left(n^2\right)
\end{array}
\][/tex]

Some known functions encountered when analyzing algorithms (Complexity category):

1. Constant: [tex]O(1)[/tex]
2. Logarithmic: [tex]O(\log n)[/tex]
3. Linear: [tex]O(n)[/tex]
4. Linearithmic: [tex]O(n \log n)[/tex]
5. Quadratic: [tex]O(n^2)[/tex]
6. Cubic: [tex]O(n^3)[/tex]
7. Exponential: [tex]O(2^n)[/tex]
8. Factorial: [tex]O(n!)[/tex]



Answer :

GinnyAnswer
Sure, let's go through the process of determining the Big O notation for the given functions step by step.

### Step-by-Step Solution

#### Step 1: Understand the given functions

Consider the two functions provided:

1. [tex]\( T_1(n) = 3n + 5 \)[/tex]
2. [tex]\( T_2(n) = 3n^2 + 4n + 2 \)[/tex]

#### Step 2: Identify the highest order term

In Big O notation, we are interested in how the function behaves as [tex]\( n \)[/tex] becomes very large. The growth rate is dominated by the term with the highest power of [tex]\( n \)[/tex]. Therefore, we need to identify the highest order term in each function.

For [tex]\( T_1(n) = 3n + 5 \)[/tex]:
- The terms are [tex]\( 3n \)[/tex] and [tex]\( 5 \)[/tex].
- The highest order term is [tex]\( 3n \)[/tex], which is linear.

For [tex]\( T_2(n) = 3n^2 + 4n + 2 \)[/tex]:
- The terms are [tex]\( 3n^2 \)[/tex], [tex]\( 4n \)[/tex], and [tex]\( 2 \)[/tex].
- The highest order term is [tex]\( 3n^2 \)[/tex], which is quadratic.

#### Step 3: Ignore coefficients and lower-order terms

In Big O notation, we are only interested in the term with the highest growth rate, and we ignore constant coefficients and lower-order terms. This simplification focuses on the term that has the most significant impact as [tex]\( n \)[/tex] grows.

For [tex]\( T_1(n) \)[/tex]:
- The highest order term [tex]\( 3n \)[/tex] can be simplified to [tex]\( n \)[/tex].
- Therefore, [tex]\( T_1(n) \)[/tex] falls under the Big O category [tex]\( O(n) \)[/tex].

For [tex]\( T_2(n) \)[/tex]:
- The highest order term [tex]\( 3n^2 \)[/tex] can be simplified to [tex]\( n^2 \)[/tex].
- Therefore, [tex]\( T_2(n) \)[/tex] falls under the Big O category [tex]\( O(n^2) \)[/tex].

#### Step 4: Summarize the results

- For [tex]\( T_1(n) = 3n + 5 \)[/tex], the Big O notation is [tex]\( O(n) \)[/tex].
- For [tex]\( T_2(n) = 3n^2 + 4n + 2 \)[/tex], the Big O notation is [tex]\( O(n^2) \)[/tex].

### Final Answer

The Big O notations for the given functions are:

1. [tex]\( T_1(n) = 3n + 5 \rightarrow O(n) \)[/tex]
2. [tex]\( T_2(n) = 3n^2 + 4n + 2 \rightarrow O(n^2) \)[/tex]

Thus, the Big O notations for the highest order terms are [tex]\( O(n) \)[/tex] and [tex]\( O(n^2) \)[/tex] respectively.

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