Answer :
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.
### 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.