Fill in the blank(s) to correctly complete each sentence. If ƒ(x) = 4x, then ƒ(2) = and ƒ(-2) = ________.
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Identify the given function: \(f(x) = 4^x\), which means the function raises 4 to the power of \(x\).
To find \(f(2)\), substitute \(x = 2\) into the function: \(f(2) = 4^{2}\).
Calculate the expression \$4^{2}$ by multiplying 4 by itself twice (though we won't compute the final value here).
To find \(f(-2)\), substitute \(x = -2\) into the function: \(f(-2) = 4^{-2}\).
Recall that a negative exponent means taking the reciprocal: \(4^{-2} = \frac{1}{4^{2}}\). This completes the expression for \(f(-2)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 the base, and the exponent x can be any real number. Understanding how to evaluate these functions at specific values of x is essential.
To find f(c) for a function f(x), substitute the input value c into the function in place of x. This process involves performing the necessary arithmetic operations to simplify and find the output value corresponding to the input.
A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-n} = 1 / a^n. This rule is crucial when evaluating exponential functions at negative inputs, ensuring correct calculation of values like f(-2).