Asked by Mucho
What is the range of this exponential function?
y = -3(1.5)^x
y = -3(1.5)^x
Answers
Answered by
Mucho
Wrong
Answered by
GPT-4o mini
To determine the range of the exponential function \( y = -3(1.5)^x \), we analyze the properties of the function:
1. **Base of the Exponential**: The base \( 1.5 \) is greater than 1, which means that as \( x \) increases, the value of \( (1.5)^x \) will increase towards positive infinity.
2. **Multiplication by -3**: Since the function is multiplied by \(-3\), this modifies the behavior of the function:
- As \( x \to -\infty \), \( (1.5)^x \) approaches 0, and consequently \( y \) approaches \( -3 \times 0 = 0 \).
- As \( x \to \infty \), \( (1.5)^x \) approaches infinity, and thus \( y \) approaches \( -3 \times \infty = -\infty \).
3. **Vertical Shift**: The negative sign in front of the \( 3 \) indicates that the entire function is reflected across the x-axis. Thus, we know that the maximum value of \( y \) approaches \( 0 \), but never actually reaches it.
Based on this analysis, we conclude:
- The maximum value of \( y \) is just below \( 0 \) (but never reaches it).
- There is no lower limit on \( y \) as it can decrease indefinitely towards \( -\infty \).
Thus, the range of the function \( y = -3(1.5)^x \) is:
\[
(-\infty, 0)
\]
1. **Base of the Exponential**: The base \( 1.5 \) is greater than 1, which means that as \( x \) increases, the value of \( (1.5)^x \) will increase towards positive infinity.
2. **Multiplication by -3**: Since the function is multiplied by \(-3\), this modifies the behavior of the function:
- As \( x \to -\infty \), \( (1.5)^x \) approaches 0, and consequently \( y \) approaches \( -3 \times 0 = 0 \).
- As \( x \to \infty \), \( (1.5)^x \) approaches infinity, and thus \( y \) approaches \( -3 \times \infty = -\infty \).
3. **Vertical Shift**: The negative sign in front of the \( 3 \) indicates that the entire function is reflected across the x-axis. Thus, we know that the maximum value of \( y \) approaches \( 0 \), but never actually reaches it.
Based on this analysis, we conclude:
- The maximum value of \( y \) is just below \( 0 \) (but never reaches it).
- There is no lower limit on \( y \) as it can decrease indefinitely towards \( -\infty \).
Thus, the range of the function \( y = -3(1.5)^x \) is:
\[
(-\infty, 0)
\]
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