Question

or
Complete the table of values for f(x)=8x2 and g(x)=3x+10.
x f(x) g(x)
1
2
3
4
Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?

Answer 1

To complete the table of values for the functions \( f(x) = 8x^2 \) and \( g(x) = 3x + 10 \), we will calculate the values for both functions at \( x = 1, 2, 3, \) and \( 4 \).

1. For \( x = 1 \):
\[ f(1) = 8(1)^2 = 8 \]
\[ g(1) = 3(1) + 10 = 3 + 10 = 13 \]

2. For \( x = 2 \):
\[ f(2) = 8(2)^2 = 8 \times 4 = 32 \]
\[ g(2) = 3(2) + 10 = 6 + 10 = 16 \]

3. For \( x = 3 \):
\[ f(3) = 8(3)^2 = 8 \times 9 = 72 \]
\[ g(3) = 3(3) + 10 = 9 + 10 = 19 \]

4. For \( x = 4 \):
\[ f(4) = 8(4)^2 = 8 \times 16 = 128 \]
\[ g(4) = 3(4) + 10 = 12 + 10 = 22 \]

Now we can summarize the results in a table:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
1 & 8 & 13 \\
2 & 32 & 16 \\
3 & 72 & 19 \\
4 & 128 & 22 \\
\hline
\end{array}
\]

Both functions \( f(x) \) and \( g(x) \) grow as \( x \) gets larger. However, since \( f(x) = 8x^2 \) is a quadratic function (and thus grows faster than a linear function), it will eventually exceed \( g(x) = 3x + 10 \) for sufficiently large values of \( x \). In our calculations, we can see that even for \( x = 2 \), \( f(x) \) starts to grow much larger than \( g(x) \), and this trend will continue as \( x \) increases. Therefore, \( f(x) \) will eventually exceed \( g(x) \).