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Sagot :
To address the problem, we need to understand how an asymptote and the range of an exponential growth function can be modified by adding or subtracting a constant. Exponential functions typically have the form \( y = a e^{bx} + c \), where \( c \) is a vertical shift.
Let’s break down the different options provided and their effects on the function:
1. A whole number constant could have been added to the exponential expression:
Suppose we have the function \( y = e^{x} + c \). Adding a constant \( c \) would shift the entire function upward by \( c \). If \( c \) was any positive number, it would raise the asymptote, but it wouldn't allow the range to include negative numbers since exponential growth functions with a positive exponent are always positive.
2. A whole number constant could have been subtracted from the exponential expression:
Consider the function \( y = e^{x} - c \). Subtracting a whole number constant \( c \) from the exponential expression translates the graph of the function downward by \( c \). If \( c \) is large enough, it will move the asymptote downward. For instance, if you have \( y = e^{x} - 5 \), the horizontal asymptote will be \( y = -5 \). This adjustment enables the range of the function to potentially include negative values when \( c \) is sufficiently large.
3. A whole number constant could have been added to the exponent:
Examining \( y = e^{x + c} \), adding a constant to the exponent translates the graph horizontally but does not affect the horizontal asymptote or make the function capable of achieving negative values, as exponential growth remains positive regardless of horizontal shifts.
4. A whole number constant could have been subtracted from the exponent:
Looking at \( y = e^{x - c} \), subtracting a constant from the exponent also translates the graph horizontally in the opposite direction but similarly does not affect the horizontal asymptote nor permits the function to achieve negative values.
Given that the function’s range needs to include negative numbers and have an asymptote of \( y = -3 \), the most plausible scenario is subtracting a whole number from the exponential expression to lower the function vertically.
Therefore, the correct option is:
A whole number constant could have been subtracted from the exponential expression.
So, the correct answer to the question is:
2: A whole number constant could have been subtracted from the exponential expression.
Let’s break down the different options provided and their effects on the function:
1. A whole number constant could have been added to the exponential expression:
Suppose we have the function \( y = e^{x} + c \). Adding a constant \( c \) would shift the entire function upward by \( c \). If \( c \) was any positive number, it would raise the asymptote, but it wouldn't allow the range to include negative numbers since exponential growth functions with a positive exponent are always positive.
2. A whole number constant could have been subtracted from the exponential expression:
Consider the function \( y = e^{x} - c \). Subtracting a whole number constant \( c \) from the exponential expression translates the graph of the function downward by \( c \). If \( c \) is large enough, it will move the asymptote downward. For instance, if you have \( y = e^{x} - 5 \), the horizontal asymptote will be \( y = -5 \). This adjustment enables the range of the function to potentially include negative values when \( c \) is sufficiently large.
3. A whole number constant could have been added to the exponent:
Examining \( y = e^{x + c} \), adding a constant to the exponent translates the graph horizontally but does not affect the horizontal asymptote or make the function capable of achieving negative values, as exponential growth remains positive regardless of horizontal shifts.
4. A whole number constant could have been subtracted from the exponent:
Looking at \( y = e^{x - c} \), subtracting a constant from the exponent also translates the graph horizontally in the opposite direction but similarly does not affect the horizontal asymptote nor permits the function to achieve negative values.
Given that the function’s range needs to include negative numbers and have an asymptote of \( y = -3 \), the most plausible scenario is subtracting a whole number from the exponential expression to lower the function vertically.
Therefore, the correct option is:
A whole number constant could have been subtracted from the exponential expression.
So, the correct answer to the question is:
2: A whole number constant could have been subtracted from the exponential expression.
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