Question

Complete the table of values for f(x)=x2+5 and g(x)=4x.
x f(x) g(x)
0
1
2
3
Both f(x) and g(x) grow as x gets larger and larger. Which function eventually exceeds the other?

Answer 1

Let's complete the table of values for the functions \( f(x) = x^2 + 5 \) and \( g(x) = 4x \).

First, we'll calculate the values for both functions at \( x = 0, 1, 2, 3 \):

1. For \( x = 0 \):
- \( f(0) = 0^2 + 5 = 5 \)
- \( g(0) = 4 \cdot 0 = 0 \)

2. For \( x = 1 \):
- \( f(1) = 1^2 + 5 = 6 \)
- \( g(1) = 4 \cdot 1 = 4 \)

3. For \( x = 2 \):
- \( f(2) = 2^2 + 5 = 9 \)
- \( g(2) = 4 \cdot 2 = 8 \)

4. For \( x = 3 \):
- \( f(3) = 3^2 + 5 = 14 \)
- \( g(3) = 4 \cdot 3 = 12 \)

Now, we can fill in the table:

| x | f(x) | g(x) |
|---|------|------|
| 0 | 5 | 0 |
| 1 | 6 | 4 |
| 2 | 9 | 8 |
| 3 | 14 | 12 |

### Analyzing Growth:
To determine which function eventually exceeds the other, consider their growth rates:

- \( f(x) = x^2 + 5 \) is a quadratic function, and its growth rate accelerates as \( x \) increases, because quadratic functions grow faster than linear functions.
- \( g(x) = 4x \) is a linear function.

Thus, as \( x \) becomes very large, \( f(x) \) will eventually exceed \( g(x) \). Specifically, the quadratic term \( x^2 \) in \( f(x) \) will dominate the linear term \( 4x \) in \( g(x) \).

In conclusion, \( f(x) \) will eventually exceed \( g(x) \) for sufficiently large values of \( x \).