**$f(x)=2x^{2}+x^{3}\log x$****$f(x)=3x^{5}+(log x)^{4}$****$f(x)=\dfrac{x^{4}+x^{2}+1}{x^{4}+1}$**

The** article aims** to find the value of the **n** for each function given to satisfy the **O(x^n)** **notation**. **Big-O** **notation represents the maximum operating time** of the algorithm. Therefore, it provides the **worst possible algorithm.** In **computer science,** big **O** notation is used to classify algorithms according to how their working time or space requirements grow as input size. In the theory of **numerical analysis**, the main notation of **O** is often used to express the obligation of the** distinction between arithmetical function and best-understood guesses;** a famous example of such a difference is the word remaining in the prime number theorem.

**Expert Answer**

**Part (a)**

The **function** is \[f(x)=2x^{2}+x^{3}\log x\]

The **property** $\log x\leq x$ **holds** when $x >0$.

\[f(x)=2x^{2}+x^{3}\log x \leq 2x^{2}+x^{4}\]

The **maximum power** of $x$ in the** expression** of the $f(x)$ is the** smallest** $n$ for which $f(x)$ is $O(x^{n})$.

\[n=4\]

When $x>2$, we have the **property** $x^{2}>x>2$.

Let’s **choose** $k=2$ first and then **choose** $x>2$.

\[|f(x)|=|2x^{2}+x^{3}\log x|\leq|2x^{2}+x^{4}|\leq |2x^{2}|+|x^{4}|\]

\[=2x^{2}+x^{4}\leq x^{4}+x^{4}\]

\[=2x^{4}\]

\[=2|x^{4}|\]

Thus, $C$ **should be at least** $2$. Let us then **choose** $C=2$.

Hence, $f(x)=O(x^{4})$ with $k=2$ and $C=2$.

**Part(b)**

The function is \[f(x)=3x^{5}+(\log x)^{4}\]

The **maximum power** of $x$ in the expression of the $f(x)$ is the **smallest** $n$ for which $f(x)$ is $O(x^{n})$.

\[n=5\]

The **property** $\log x\leq x$ holds when $x, 0$.

When $x>1$, we have the **property** $x^{4}<x^{5}$.

Let’s **choose** $k=1$ first and then **choose** $x>1$.

\[|f(x)|=|3x^{5}+(\log x)^{4}|\leq|3x^{5}|+|(\log x)^{4}|\]

\[=3x^{5}+(\log x)^{4}\leq 3x^{5}+x^{4}\]

\[=4x^{5}\]

\[=4|x^{5}|\]

Thus, $C$ **should be at least** $4$. Let us then choose $C=4$.

**The Big $O$ notation**, $f(x)=O(x^{5})$ with $k=1$ and $C=4$.

**Part(c)**

The** function** is \[f(x)=\frac{x^{4}+x^{2}+1}{x^{4}+1}\]

Let’s determine the quotient of the **reminder using long division.**

The **quotient** is $1$ with **reminder** $x^{2}$.

**Rewrite the given fraction**

\[f(x)=\frac{x^{4}+x^{2}+1}{x^{4}+1}\]

\[f(x)=1+\frac{x^{2}+1}{x^{4}+1}\]

The **maximum power** of $x$ in the **expression** of the $f(x)$ is the **smallest** $n$ for which $f(x)$ is $O(x^{n})$.

\[n=0\]

Let’s **choose** $k=0$ first and then **choose** $x>0$.

\[|f(x)|=|1+\frac{x^{2}+1}{x^{4}+1}|\leq |1|+|\frac{x^{2}}{x^{4}+1}|\]

\[|f(x)|=1+\frac{x^{2}}{x^{4}+1}\leq 1+1\]

\[=3x^{5}+(\log x)^{4}\leq 3x^{5}+x^{4}<2\]

\[=2.1\]

\[=2|x^{o}|\]

Thus, $C$ **should be at least** $2$. Let us then choose $C=2$.

**Numerical Result**

**-$f(x)=2x^{2}+x^{3}\log x$**

**The Big $O$ notation,** $f(x)=O(x^{4})$ with $k=2$ and $C=2$.

**-$f(x)=3x^{5}+(log x)^{4}$**

**T****he Big $O$ notation**, $f(x)=O(x^{5})$ with $k=1$ and $C=4$.

**-$f(x)=\dfrac{x^{4}+x^{2}+1}{x^{4}+1}$**

**The Big $O$ notation,** $f(x)=O(x^{0})=O(1)$ with $k=0$ and $C=2$.

**Example**

**Determine the least integer $n$ such that $f(x)$ is $O(x^{n}) for the following functions.**

**-$f(x)=2x^{2}+x^{4}\log x$**

**Solution**

The **function** is \[f(x)=2x^{2}+x^{4}\log x\]

The** property** $\log x\leq x$ holds when $x >0$.

\[f(x)=2x^{2}+x^{4}\log x \leq 2x^{2}+x^{5}\]

The **highest power** of $x$ in the **expression** of the $f(x)$ is the **smallest** $n$ for which $f(x)$ is $O(x^{n})$.

\[n=5\]

When $x>2$, we have the **property** $x^{2}>x>2$.

Let’s **choose** $k=2$ first and then choose $x>2$.

\[|f(x)|=|2x^{2}+x^{4}\log x|\leq|2x^{2}+x^{5}|\leq |2x^{2}|+|x^{5}|\]

\[=2x^{2}+x^{5}\leq x^{5}+x^{5}\]

\[=2x^{5}\]

\[=2|x^{5}|\]

Thus, $C$** should be at least** $2$. Let us then** choose** $C=2$.