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Function evaluation involves finding the value of a given function for specific values of its variable. This process is essential in determining how the function behaves at different points. Usually, a function is expressed in terms of a variable—such as the function \(f(x) = 3^x\). Here, \(x\) is the variable, and 3 is the base of the exponent.To evaluate this function at a particular \(x\) value, simply substitute the chosen \(x\) value into the function. For instance, if you are asked to find \(f(-2)\), substitute \(-2\) for \(x\) in the expression \(3^x\). This results in \(3^{-2}\), which you can then simplify using your knowledge of exponents.It's what we do when we find specific outputs of a function, helping us see how a function changes for different inputs.

Negative exponents might seem tricky at first, but they're not too hard to understand. The key property to remember is: when you have \(a^{-b}\), it equals \(\frac{1}{a^b}\). This means you're taking the reciprocal of the base raised to the positive version of the exponent.For example, let's consider the expression \(3^{-1}\). According to the property, this is equivalent to \(\frac{1}{3^1}\), which simplifies to \(\frac{1}{3}\). Similarly, \(3^{-2}\) would be \(\frac{1}{3^2} = \frac{1}{9}\).This concept turns the base of the exponent into a fraction, which is a handy way to simplify expressions without negative powers.

Understanding the powers of a number is crucial for evaluating exponential functions. This involves raising a number called the base, to a certain power, also known as the exponent.When you see something like \(3^2\), it means you're multiplying 3 by itself once: \(3 \times 3 = 9\). The concepts become more intuitive when the exponent is zero or one.When any non-zero number is raised to the power of zero, the result is always 1. So, for example, \(3^0 = 1\). If the exponent is one, then the number remains unchanged. Thus, \(3^1 = 3\).Keep these rules in check, and you will master working with any power-related problems easily!

The substitution method is a straightforward way to evaluate functions. It involves replacing a variable in the equation with a specific value to find the result. Using our example function, \(f(x) = 3^x\), and evaluating it at different \(x\) values helps us understand how the function behaves.### Example: Evaluating Function at Different Points- To find \(f(-2)\), substitute \(-2\) into the function, giving \(3^{-2}\), simplify using negative exponents to get \(\frac{1}{9}\).- For \(f(-1)\), substitute \(-1\), resulting in \(3^{-1} = \frac{1}{3}\).- Determine \(f(0)\) by substituting \(0\), giving \(3^0 = 1\).- Substituting \(1\) yields \(f(1) = 3^1 = 3\).- Finally, substituting \(2\) gives \(f(2) = 3^2 = 9\).This method is powerful for pinpointing values on a function and is widely used for solving equations where functions are involved.