Question

Complete the table of values for f(x)=4x2–6 and g(x)=5x–2.
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 \( f(x) = 4x^2 - 6 \) and \( g(x) = 5x - 2 \), we will calculate the values of \( f(x) \) and \( g(x) \) for \( x = 1, 2, 3, \) and \( 4 \).

### Calculating \( f(x) \) and \( g(x) \):

1. **For \( x = 1 \):**
- \( f(1) = 4(1^2) - 6 = 4(1) - 6 = 4 - 6 = -2 \)
- \( g(1) = 5(1) - 2 = 5 - 2 = 3 \)

2. **For \( x = 2 \):**
- \( f(2) = 4(2^2) - 6 = 4(4) - 6 = 16 - 6 = 10 \)
- \( g(2) = 5(2) - 2 = 10 - 2 = 8 \)

3. **For \( x = 3 \):**
- \( f(3) = 4(3^2) - 6 = 4(9) - 6 = 36 - 6 = 30 \)
- \( g(3) = 5(3) - 2 = 15 - 2 = 13 \)

4. **For \( x = 4 \):**
- \( f(4) = 4(4^2) - 6 = 4(16) - 6 = 64 - 6 = 58 \)
- \( g(4) = 5(4) - 2 = 20 - 2 = 18 \)

### Completed Table:
| \( x \) | \( f(x) \) | \( g(x) \) |
|---------|------------|------------|
| 1 | -2 | 3 |
| 2 | 10 | 8 |
| 3 | 30 | 13 |
| 4 | 58 | 18 |

### Analysis:
As the values show, for \( x = 1 \), \( g(x) \) is greater than \( f(x) \). At \( x = 2 \), \( f(x) \) exceeds \( g(x) \) and continues to grow faster than \( g(x) \).

### Conclusion:
Eventually, the quadratic function \( f(x) = 4x^2 - 6 \) will always exceed the linear function \( g(x) = 5x - 2 \) as \( x \) increases, because the quadratic function grows faster than the linear function for large values of \( x \).