Verified Answer :

  • The function is defined as \(f(x) = 3^x\).
  • Substituting \(x = -3\) gives \(f(-3) = 3^{-3}\).
  • Using the property of negative exponents, \(3^{-3} = \frac{1}{3^3}\).
  • Since \(3^3 = 27\), the final result is \(\boxed{\frac{1}{27}}\).

Explanation

  1. Understanding the Problem and Function Definition

The function is given by \(f(x) = 3^x\). We are asked to evaluate this function at \(x = -3\). To do this, we substitute the value into the expression: \(f(-3) = 3^{-3}\).

  1. Applying the Negative Exponent Rule

Recall the rule for negative exponents, which states that for any non-zero real number \(a\) and positive integer \(n\), the expression \(a^{-n}\) is equivalent to \(\frac{1}{a^n}\). Applying this property to our expression, we have:
\[3^{-3} = \frac{1}{3^3}\]

  1. Calculating the Final Result

Now, calculate the value in the denominator: \(3^3 = 3 \times 3 \times 3 = 27\). Therefore, substituting this back into our fraction gives:
\[3^{-3} = \frac{1}{27}\]
Thus, the value of the function at \(x = -3\) is \(\frac{1}{27}\).

Examples

In real-world applications like financial modeling, negative exponents appear when calculating the present value of money that will be received in the future. For example, if an investment grows by a factor of 3 each year, the value of an initial investment \(V_0\) after \(n\) years is \(V_n = V_0 \cdot 3^n\). To find the value of an amount expected 3 years in the past (at an equivalent growth rate), we use a negative exponent: \(V_{-3} = V_0 \cdot 3^{-3} = V_0 \cdot \frac{1}{27}\), which shows how growth factors apply inversely when looking backward in time.

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