The function \( f(x) = 3^x \) is an exponential function where the base \( 3 \) is greater than \( 1 \). Exponential functions of this type have specific characteristics regarding their range.
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Behavior of the function: As \( x \) approaches negative infinity (\( x \to -\infty \)), \( 3^x \) approaches \( 0 \) but never actually reaches it (i.e., \( 3^x \to 0^+ \)). As \( x \) approaches positive infinity (\( x \to \infty \)), \( 3^x \) increases without bound (i.e., \( 3^x \to \infty \)).
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Value at \( x = 0 \): At \( x = 0 \), \( f(0) = 3^0 = 1 \).
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Conclusion about the range: The function \( f(x) = 3^x \) can take all positive values greater than \( 0 \). Therefore, the range of \( f(x) \) is:
\[ (0, \infty) \]
This means \( f(x) \) never reaches \( 0 \) and can become infinitely large. Hence, the range is all real numbers greater than \( 0 \).
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