To determine the range of the exponential function \( y = -3(1.5)^x \), we analyze the properties of the function:
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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.
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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 \( -3 \times \infty = -\infty \).
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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:
\[ (-\infty, 0) \]