The graph provided compares different runtime functions. The curves for $T(n) = 2 \log n + 1$, $f_A(n) = 2$, and $f_B(n) = 2n + 1$ are plotted. Based on the descriptions in the second image and the data from the graph, I can analyze the functions as follows:

1. $f_C(n) = 2n^{0.5} + 1$ grows faster than $T(n) = 2 \log n + 1$ as $n$ increases:

   • This is true. The curve for $f_C(n) = 2n^{0.5} + 1$ grows faster than $T(n) = 2 \log n + 1$, which shows a logarithmic increase. In comparison, $f_C(n)$ exhibits a growth rate faster than logarithmic as $n$ increases.

2. At around $n = 10000$ and beyond, $f_B(n) = 2n + 1$ has a (way) slower runtime than all of the three functions $T(n), f_A(n),$ and $f_C(n)$:

   • This is false. From the graph, $f_B(n) = 2n + 1$ increases linearly, and it clearly grows faster than the constant $f_A(n) = 2$ and logarithmic $T(n) = 2 \log n + 1$, and slower than $f_C(n)$. However, it does not have a slower runtime than all of the functions.

3. $f_C(n) = 2n^{0.5} + 1$ grows faster than $f_B(n) = 2n + 1$ as $n$ increases:

   • This is false. $f_B(n) = 2n + 1$ grows linearly, while $f_C(n) = 2n^{0.5} + 1$ grows at a rate slower than linear, so it does not grow faster than $f_B(n)$.

4. $T(n) = 2n + 1$ will have the same value as $f_A(n) = 2$ somewhere within the interval $n \in [2, \infty)$:

   • This is true. The function $T(n) = 2n + 1$ is a linear function, and since $f_A(n) = 2$ is a constant, they will indeed intersect at some point in the interval as the values of $T(n)$ and $f_A(n)$ converge.

Based on this analysis, the two true statements are:

• $f_C(n) = 2n^{0.5} + 1$ grows faster than $T(n) = 2 \log n + 1$ as $n$ increases.
• $T(n) = 2n + 1$ will have the same value as $f_A(n) = 2$ somewhere within the interval $n \in [2, \infty)$.

Would you like further explanations or assistance with other concepts related to these functions?

Here are some follow-up questions you might find useful:

1. What is the exact point where $T(n) = 2n + 1$ intersects $f_A(n) = 2$?
2. How can we determine the growth rate of a function from its graph?
3. What are the practical applications of logarithmic and linear growth in algorithms?
4. How does the addition of $n^{0.5}$ in $f_C(n)$ affect its runtime complexity compared to linear growth?
5. Can you describe how logarithmic growth compares to polynomial growth in terms of computational cost?

Tip: To compare growth rates, you can often use big-O notation to analyze how different functions scale with large inputs.