Question
Consider the two exponential functions.
Function Function
Which two statements are true?
Responses
As increases, increases by a greater percentage than .
As x increases, f ( x ) increases by a greater percentage than g ( x ) .
As increases, increases by a lesser percentage than .
As x increases, f ( x ) increases by a lesser percentage than g ( x ) .
As increases, increases by the same percentage as .
As x increases, f ( x ) increases by the same percentage as g ( x ) .
The -intercept of is greater than the -intercept of .
The y -intercept of f ( x ) is greater than the y -intercept of g ( x ) .
The -intercept of is less than the -intercept of .
The y -intercept of f ( x ) is less than the y -intercept of g ( x ) .
Function Function
Which two statements are true?
Responses
As increases, increases by a greater percentage than .
As x increases, f ( x ) increases by a greater percentage than g ( x ) .
As increases, increases by a lesser percentage than .
As x increases, f ( x ) increases by a lesser percentage than g ( x ) .
As increases, increases by the same percentage as .
As x increases, f ( x ) increases by the same percentage as g ( x ) .
The -intercept of is greater than the -intercept of .
The y -intercept of f ( x ) is greater than the y -intercept of g ( x ) .
The -intercept of is less than the -intercept of .
The y -intercept of f ( x ) is less than the y -intercept of g ( x ) .
Answers
Answered by
GPT-4o mini
To analyze the two exponential functions, we need to clarify a few points:
1. **Nature of Exponential Functions**: Exponential functions generally take the form \( f(x) = a \cdot b^x \) where \( b > 0 \). The base \( b \) determines the growth rate; a higher base results in a steeper increase.
2. **Percentage Increase**: For two exponential functions \( f(x) = a \cdot b^x \) and \( g(x) = c \cdot d^x \):
- As \( x \) increases, the percentage increase in \( f(x) \) compared to \( g(x) \) depends largely on the bases \( b \) and \( d \).
- If \( b > d \), then \( f(x) \) increases by a greater percentage than \( g(x) \) as \( x \) tends towards infinity.
3. **Y-Intercepts**: The y-intercept of \( f(x) \) is given by \( f(0) = a \) and for \( g(x) \) it is \( g(0) = c \). Therefore, one can compare the y-intercepts directly based on the coefficients \( a \) and \( c \).
Now, without specific values for the functions, we can state the following:
1. **As \( x \) increases, \( f(x) \) increases by a greater percentage than \( g(x) \)** if the base of \( f \) is greater than the base of \( g \).
2. **The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \)** if \( a > c \), and conversely, it is less if \( a < c \).
### Conclusion:
To select true statements without specific functions, you would typically rely on the properties outlined above. However, based on a standard understanding, the following statements could be inferred as true under conditions where the higher base leads to greater growth, and the y-intercepts are directly related to the coefficients:
- **As \( x \) increases, \( f(x) \) increases by a greater percentage than \( g(x) \) (if the base of \( f \) is greater).**
- **The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \) (if \( a > c \)).**
If specific function values or conditions are provided, you could confirm or adjust these statements accordingly.
1. **Nature of Exponential Functions**: Exponential functions generally take the form \( f(x) = a \cdot b^x \) where \( b > 0 \). The base \( b \) determines the growth rate; a higher base results in a steeper increase.
2. **Percentage Increase**: For two exponential functions \( f(x) = a \cdot b^x \) and \( g(x) = c \cdot d^x \):
- As \( x \) increases, the percentage increase in \( f(x) \) compared to \( g(x) \) depends largely on the bases \( b \) and \( d \).
- If \( b > d \), then \( f(x) \) increases by a greater percentage than \( g(x) \) as \( x \) tends towards infinity.
3. **Y-Intercepts**: The y-intercept of \( f(x) \) is given by \( f(0) = a \) and for \( g(x) \) it is \( g(0) = c \). Therefore, one can compare the y-intercepts directly based on the coefficients \( a \) and \( c \).
Now, without specific values for the functions, we can state the following:
1. **As \( x \) increases, \( f(x) \) increases by a greater percentage than \( g(x) \)** if the base of \( f \) is greater than the base of \( g \).
2. **The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \)** if \( a > c \), and conversely, it is less if \( a < c \).
### Conclusion:
To select true statements without specific functions, you would typically rely on the properties outlined above. However, based on a standard understanding, the following statements could be inferred as true under conditions where the higher base leads to greater growth, and the y-intercepts are directly related to the coefficients:
- **As \( x \) increases, \( f(x) \) increases by a greater percentage than \( g(x) \) (if the base of \( f \) is greater).**
- **The y-intercept of \( f(x) \) is greater than the y-intercept of \( g(x) \) (if \( a > c \)).**
If specific function values or conditions are provided, you could confirm or adjust these statements accordingly.