1. **State the problem:** We need to complete the table of values for the functions $f(x) = 4^x - 3$ and $g(x) = 4x^2 + 2$ at $x = 1, 2, 3, 4$. 2. **Recall the formulas:** - For $f(x)$: $$f(x) = 4^x - 3$$ - For $g(x)$: $$g(x) = 4x^2 + 2$$ 3. **Calculate $f(x)$ values:** - At $x=1$: $$f(1) = 4^1 - 3 = 4 - 3 = 1$$ - At $x=2$: $$f(2) = 4^2 - 3 = 16 - 3 = 13$$ - At $x=3$: $$f(3) = 4^3 - 3 = 64 - 3 = 61$$ - At $x=4$: $$f(4) = 4^4 - 3 = 256 - 3 = 253$$ 4. **Calculate $g(x)$ values:** - At $x=1$: $$g(1) = 4(1)^2 + 2 = 4 + 2 = 6$$ - At $x=2$: $$g(2) = 4(2)^2 + 2 = 4 \times 4 + 2 = 16 + 2 = 18$$ - At $x=3$: $$g(3) = 4(3)^2 + 2 = 4 \times 9 + 2 = 36 + 2 = 38$$ - At $x=4$: $$g(4) = 4(4)^2 + 2 = 4 \times 16 + 2 = 64 + 2 = 66$$ 5. **Complete the table:** | x | f(x) | g(x) | |---|------|------| | 1 | 1 | 6 | | 2 | 13 | 18 | | 3 | 61 | 38 | | 4 | 253 | 66 | 6. **Compare growth:** - $f(x) = 4^x - 3$ is an exponential function. - $g(x) = 4x^2 + 2$ is a quadratic function. 7. **Which function eventually exceeds the other?** - Exponential functions grow faster than polynomial functions as $x$ becomes very large. - Therefore, $f(x)$ will eventually exceed $g(x)$ for sufficiently large $x$. **Final answer:** - Table values as above. - $f(x)$ eventually exceeds $g(x)$ as $x$ grows larger.