Question
Which function grows at the fastest rate for increasing values of x?
Responses
f(x)=8.5x+3
f begin argument x end argument equals 8.5 x plus 3
h(x)=9.5x2
h begin argument x end argument equals 9.5 x squared
g(x)=2.5x
Responses
f(x)=8.5x+3
f begin argument x end argument equals 8.5 x plus 3
h(x)=9.5x2
h begin argument x end argument equals 9.5 x squared
g(x)=2.5x
Answers
GPT-4o mini
To determine which function grows at the fastest rate for increasing values of \( x \), we can consider the forms of the functions given:
1. \( f(x) = 8.5x + 3 \) - This is a linear function.
2. \( g(x) = 2.5x \) - This is also a linear function.
3. \( h(x) = 9.5x^2 \) - This is a quadratic function.
When comparing the growth rates of these functions:
- Linear functions like \( f(x) \) and \( g(x) \) grow at a constant rate. The growth rate is determined by the coefficient of \( x \), which are \( 8.5 \) and \( 2.5 \), respectively. Thus, \( f(x) \) grows faster than \( g(x) \) as \( x \) increases.
- Quadratic functions like \( h(x) \) grow at a rate that increases as \( x \) increases. Specifically, the growth rate of \( h(x) \) is proportional to \( x^2 \), which means as \( x \) becomes large, \( h(x) \) will outpace both linear functions.
As a result, among the functions provided, the one that grows at the fastest rate for increasing values of \( x \) is:
**\( h(x) = 9.5x^2 \)**
1. \( f(x) = 8.5x + 3 \) - This is a linear function.
2. \( g(x) = 2.5x \) - This is also a linear function.
3. \( h(x) = 9.5x^2 \) - This is a quadratic function.
When comparing the growth rates of these functions:
- Linear functions like \( f(x) \) and \( g(x) \) grow at a constant rate. The growth rate is determined by the coefficient of \( x \), which are \( 8.5 \) and \( 2.5 \), respectively. Thus, \( f(x) \) grows faster than \( g(x) \) as \( x \) increases.
- Quadratic functions like \( h(x) \) grow at a rate that increases as \( x \) increases. Specifically, the growth rate of \( h(x) \) is proportional to \( x^2 \), which means as \( x \) becomes large, \( h(x) \) will outpace both linear functions.
As a result, among the functions provided, the one that grows at the fastest rate for increasing values of \( x \) is:
**\( h(x) = 9.5x^2 \)**