To determine the range of the exponential function \( f(x) = 3\left(\frac{1}{2}\right)^x - 6 \), we will analyze the behavior of the function as \( x \) varies.
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Exponential Component: The term \( \left(\frac{1}{2}\right)^x \) is an exponential function that decreases as \( x \) increases. Its range is \( (0, \infty) \). Specifically:
- As \( x \to -\infty \), \( \left(\frac{1}{2}\right)^x \to \infty \).
- As \( x \to \infty \), \( \left(\frac{1}{2}\right)^x \to 0 \).
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Applying the Coefficient: The function \( f(x) = 3\left(\frac{1}{2}\right)^x \) has the same behavior as \( \left(\frac{1}{2}\right)^x \) but is scaled by a factor of 3:
- As \( x \to -\infty \), \( 3\left(\frac{1}{2}\right)^x \to \infty \).
- As \( x \to \infty \), \( 3\left(\frac{1}{2}\right)^x \to 0 \).
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Subtracting 6: Now we need to subtract 6 from \( 3\left(\frac{1}{2}\right)^x \):
- When \( x \to -\infty\), \( f(x) \to \infty - 6 = \infty \).
- When \( x \to \infty\), \( f(x) \to 0 - 6 = -6 \).
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Conclusion on Range: Thus, as \( x \) becomes larger, \( f(x) \) approaches -6 but never actually reaches it, and as \( x \) becomes smaller, \( f(x) \) can grow large without bound.
Therefore, the range of the function \( f(x) = 3\left(\frac{1}{2}\right)^x - 6 \) is:
\[ \text{Range} = (-6, \infty) \]
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