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 \( x = 0, 1, 2, 3 \):
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For \( x = 0 \):
- \( f(0) = 0^2 + 5 = 5 \)
- \( g(0) = 4 \cdot 0 = 0 \)
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For \( x = 1 \):
- \( f(1) = 1^2 + 5 = 6 \)
- \( g(1) = 4 \cdot 1 = 4 \)
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For \( x = 2 \):
- \( f(2) = 2^2 + 5 = 9 \)
- \( g(2) = 4 \cdot 2 = 8 \)
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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 \( x \).