Question
Each of these functions grows as x gets larger and larger. Which function eventually exceeds the others?
Responses
g(x)=2x+1
g of x is equal to 2 to the x th power plus 1
f(x)=2x+1
f of x is equal to 2 x plus 1
h(x)=x2+1
Responses
g(x)=2x+1
g of x is equal to 2 to the x th power plus 1
f(x)=2x+1
f of x is equal to 2 x plus 1
h(x)=x2+1
Answers
GPT-4o mini
To determine which function eventually exceeds the others as \( x \) gets larger, we can analyze the growth rates of each function.
1. **\( g(x) = 2^x + 1 \)**: This is an exponential function. Exponential functions grow much faster than polynomial functions as \( x \) approaches infinity.
2. **\( f(x) = 2x + 1 \)**: This is a linear function. Linear functions have a constant rate of growth, specifically a slope of 2. They grow slower than exponential functions and polynomial functions (with a degree greater than 1) when \( x \) becomes large.
3. **\( h(x) = x^2 + 1 \)**: This is a quadratic function. Quadratic functions grow faster than linear functions but slower than exponential functions as \( x \) gets large.
Now, let's compare the growth rates:
- For large values of \( x \):
- **Exponential vs. Polynomial**: \( g(x) = 2^x + 1 \) will eventually exceed \( h(x) \) (the quadratic) and \( f(x) \) (the linear function).
- **Quadratic vs. Linear**: \( h(x) = x^2 + 1 \) will eventually exceed \( f(x) = 2x + 1 \) since the growth of \( x^2 \) will outpace the linear term \( 2x \) as \( x \) increases.
Thus, as \( x \) gets larger and larger:
- \( f(x) \) will be the smallest.
- \( h(x) \) will grow faster than \( f(x) \) but slower than \( g(x) \).
- \( g(x) \) (the exponential function) will eventually exceed both \( f(x) \) and \( h(x) \).
In conclusion, **the function that eventually exceeds the others as \( x \) becomes large is \( g(x) = 2^x + 1 \)**.
1. **\( g(x) = 2^x + 1 \)**: This is an exponential function. Exponential functions grow much faster than polynomial functions as \( x \) approaches infinity.
2. **\( f(x) = 2x + 1 \)**: This is a linear function. Linear functions have a constant rate of growth, specifically a slope of 2. They grow slower than exponential functions and polynomial functions (with a degree greater than 1) when \( x \) becomes large.
3. **\( h(x) = x^2 + 1 \)**: This is a quadratic function. Quadratic functions grow faster than linear functions but slower than exponential functions as \( x \) gets large.
Now, let's compare the growth rates:
- For large values of \( x \):
- **Exponential vs. Polynomial**: \( g(x) = 2^x + 1 \) will eventually exceed \( h(x) \) (the quadratic) and \( f(x) \) (the linear function).
- **Quadratic vs. Linear**: \( h(x) = x^2 + 1 \) will eventually exceed \( f(x) = 2x + 1 \) since the growth of \( x^2 \) will outpace the linear term \( 2x \) as \( x \) increases.
Thus, as \( x \) gets larger and larger:
- \( f(x) \) will be the smallest.
- \( h(x) \) will grow faster than \( f(x) \) but slower than \( g(x) \).
- \( g(x) \) (the exponential function) will eventually exceed both \( f(x) \) and \( h(x) \).
In conclusion, **the function that eventually exceeds the others as \( x \) becomes large is \( g(x) = 2^x + 1 \)**.