1. Determine whether each of these functions is O(x).a) f (x) = 10 b) f (x) = 3x + 7 c) f (x) = x2 + x + 1 d) f (x) = 5 log x

1. Break Down the Problem

We need to determine whether each of the given functions $f(x)$ is $O(x)$. This involves checking if there are constants $C$ and $n_0$ such that for all $x \geq n_0$,

$|f(x)| \leq C |x|$

2. Relevant Concepts

The notation $O(g(x))$ represents the set of functions that grow no faster than $g(x)$ up to constant factors. We will analyze each function in relation to $x$.

3. Analysis and Detail

a) $f(x) = 10$

• For small values of $x$ (e.g., $x = 1$), $10 \leq C |x|$ is valid for $C = 10$ and $n_0 = 1$.
• As $x$ increases, $10$ remains constant, while $x$ grows. Thus, for all $x \geq 1$, $10 \leq Cx$ holds true.

Conclusion: $f(x) = 10$ is $O(x)$.

b) $f(x) = 3x + 7$

• For large $x$, $|3x + 7| \approx |3x|$.
• We can choose $C = 4$ (since $3x + 7 \leq 4x$ for $x \geq 1$) and $n_0 = 1$.

Conclusion: $f(x) = 3x + 7$ is $O(x)$.

c) $f(x) = x^2 + x + 1$

• The dominant term is $x^2$.
• We analyze: For large $x$, $|x^2 + x + 1| \approx |x^2|$.
• We compare it to $Cx$: $x^2$ grows faster than $Cx$ for any constant $C$ when $x$ is large. Therefore, no constants exist.

Conclusion: $f(x) = x^2 + x + 1$ is not $O(x)$.

d) $f(x) = 5 \log x$

• The growth rate of $\log x$ is slower than any linear function $x$.
• For sufficiently large $x$, $5 \log x \leq C |x|$ where $C$ can be chosen (e.g., $C=1$ works for $n_0$ chosen around $e^C$).

Conclusion: $f(x) = 5 \log x$ is $O(x)$.

4. Verify and Summarize

The functions and their classifications are:

1. $f(x) = 10$ is $O(x)$.
2. $f(x) = 3x + 7$ is $O(x)$.
3. $f(x) = x^2 + x + 1$ is not $O(x)$.
4. $f(x) = 5 \log x$ is $O(x)$.

Final Answer

• $f(x) = 10$: Yes, $O(x)$
• $f(x) = 3x + 7$: Yes, $O(x)$
• $f(x) = x^2 + x + 1$: No, not $O(x)$
• $f(x) = 5 \log x$: Yes, $O(x)$