Question

Each of these functions grows as x gets larger and larger. Which function eventually exceeds the others?
f(x) =3^x
g(x) = 5x^2 -8
h(x) = 8x

Answer 1

To determine which function grows the fastest as \( x \) approaches infinity, we can compare their growth rates.

Let's consider each function:

1. **Exponential Function**: \( f(x) = 3^x \)
2. **Polynomial Function**: \( g(x) = 5x^2 - 8 \)
3. **Linear Function**: \( h(x) = 8x \)

### Comparing Growth Rates

1. **Exponential vs. Polynomial**:
- Exponential functions (like \( 3^x \)) grow faster than any polynomial function (like \( 5x^2 \)) for sufficiently large \( x \). This means that \( f(x) = 3^x \) will eventually exceed \( g(x) = 5x^2 - 8 \).

2. **Exponential vs. Linear**:
- Similarly, exponential functions outpace linear functions (like \( 8x \)) as \( x \) becomes very large. Thus, \( f(x) = 3^x \) will eventually exceed \( h(x) = 8x \).

3. **Comparing Polynomial and Linear**:
- Polynomials grow faster than linear functions as well. For large \( x \), \( g(x) = 5x^2 - 8 \) will exceed \( h(x) = 8x \).

### Conclusion
- As \( x \) becomes very large:
- \( 3^x \) (exponential function) will eventually exceed both \( 5x^2 - 8 \) (polynomial function) and \( 8x \) (linear function).
- From the two comparisons, \( 5x^2 - 8 \) will eventually exceed \( 8x \).

Thus, the function that eventually exceeds the others as \( x \to \infty \) is:

\[
\text{f(x) = 3}^x
\]