Question
Each of these functions grows as x gets larger and larger. Which function eventually exceeds the others?
f(x)=8x+6
g(x)=4^ x-3
h(x)=3x^2-4x-6
f(x)=8x+6
g(x)=4^ x-3
h(x)=3x^2-4x-6
Answers
GPT-4o mini
To determine which of the functions \( f(x) = 8x + 6 \), \( g(x) = 4^x - 3 \), and \( h(x) = 3x^2 - 4x - 6 \) eventually exceeds the others as \( x \) gets larger, we can analyze their growth rates.
1. **Linear Function \( f(x) = 8x + 6 \)**:
- This is a linear function. The growth rate is constant and linear.
2. **Exponential Function \( g(x) = 4^x - 3 \)**:
- This is an exponential function. Exponential functions grow much faster than linear functions (and polynomial functions) as \( x \) increases.
3. **Quadratic Function \( h(x) = 3x^2 - 4x - 6 \)**:
- This is a quadratic function. Although the quadratic function grows faster than a linear function for large \( x \), it grows slower than an exponential function.
Now, let's directly compare the growth of these functions as \( x \) approaches infinity:
- As \( x \to \infty \), the linear term \( 8x \) in \( f(x) \) grows slower than the quadratic term \( 3x^2 \) in \( h(x) \), so \( h(x) \) will eventually exceed \( f(x) \) for large \( x \).
- The quadratic function \( h(x) \) grows faster than the linear function \( f(x) \), but slower than the exponential function \( g(x) \).
- As \( x \to \infty \), the function \( g(x) = 4^x - 3 \) will eventually grow faster than both \( f(x) \) and \( h(x) \) because exponential growth outpaces linear and polynomial growth.
### Conclusion
As \( x \) becomes very large, the function \( g(x) = 4^x - 3 \) will eventually exceed both \( f(x) \) and \( h(x) \). Thus, the answer is that **\( g(x) \) eventually exceeds the others.**
1. **Linear Function \( f(x) = 8x + 6 \)**:
- This is a linear function. The growth rate is constant and linear.
2. **Exponential Function \( g(x) = 4^x - 3 \)**:
- This is an exponential function. Exponential functions grow much faster than linear functions (and polynomial functions) as \( x \) increases.
3. **Quadratic Function \( h(x) = 3x^2 - 4x - 6 \)**:
- This is a quadratic function. Although the quadratic function grows faster than a linear function for large \( x \), it grows slower than an exponential function.
Now, let's directly compare the growth of these functions as \( x \) approaches infinity:
- As \( x \to \infty \), the linear term \( 8x \) in \( f(x) \) grows slower than the quadratic term \( 3x^2 \) in \( h(x) \), so \( h(x) \) will eventually exceed \( f(x) \) for large \( x \).
- The quadratic function \( h(x) \) grows faster than the linear function \( f(x) \), but slower than the exponential function \( g(x) \).
- As \( x \to \infty \), the function \( g(x) = 4^x - 3 \) will eventually grow faster than both \( f(x) \) and \( h(x) \) because exponential growth outpaces linear and polynomial growth.
### Conclusion
As \( x \) becomes very large, the function \( g(x) = 4^x - 3 \) will eventually exceed both \( f(x) \) and \( h(x) \). Thus, the answer is that **\( g(x) \) eventually exceeds the others.**