Concept Check. If ƒ(x) = a^x and ƒ(3) = 27, determine each function value. ƒ(-1)

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Identify the given function: \(f(x) = a^{x}\), where \(a\) is the base we need to find.

Use the given information \(f(3) = 27\) to set up the equation: \(a^{3} = 27\).

Solve for \(a\) by taking the cube root of both sides: \(a = \sqrt[3]{27}\).

Once you find the value of \(a\), substitute it back into the function to find \(f(-1)\): \(f(-1) = a^{-1}\).

Recall that \(a^{-1} = \frac{1}{a}\), so express \(f(-1)\) as \(\frac{1}{a}\) using the value of \(a\) found.

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주요 개념

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Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant. The function grows or decays depending on whether a is greater than or less than 1. Understanding this form is essential to evaluate the function at different values of x.

Solving for the Base in an Exponential Function

Given a function value like f(3) = 27, you can find the base a by solving the equation a^3 = 27. This involves recognizing that 27 is a power of 3, so a can be determined by taking the cube root of 27, which is crucial for evaluating the function at other points.

Evaluating the Function at Negative Exponents

To find f(-1) when f(x) = a^x, use the property of exponents that a^{-1} = 1/a. This means evaluating the function at a negative exponent involves taking the reciprocal of the base, which is important for correctly determining function values for negative inputs.