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Function evaluation is the process of finding the output of a function at a specific input value. To evaluate a function, follow these steps:

Identify the function you need to work with. In the exercise given, the function is \( g(x) = 2^{1-x} \).

Substitute the given value of the variable into the function. For our case, we substitute \( x = -2 \) into \( g(x) \), so we have \( g(-2) \).

Simplify the expression inside the function after substitution. For \( g(-2) \), this leads to \( 2^{1-(-2)} \), which simplifies to \( 2^3 \).

Calculate the result to find the function's value at that point. Hence, \( 2^3 = 8 \). This means that \( g(-2) = 8 \).

These steps help you systematically evaluate any function.

Understanding negative exponents is crucial for simplifying expressions with powers. A negative exponent indicates the reciprocal of the base raised to the positive exponent. For instance, \( a^{-b} \) is the same as \( \frac{1}{a^b} \).

Here's a breakdown:

• \( 2^{-1} = \frac{1}{2} \)
• \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \)
• \( (\frac{1}{4})^{-2} = 4^2 = 16 \)

Knowing how to convert negative exponents to positive ones simplifies calculations and helps when working with more complex expressions. For example, in the function \( g(x) \), if we encounter a negative exponent during evaluation, we can convert it to reciprocal form for easier computation.

Simplifying expressions involves reducing them to their simplest form. This can include combining like terms, factoring, and reducing fractions. Simplifying makes it easier to evaluate and understand mathematical functions.

Steps for simplifying expressions:

• Combine like terms (terms with the same variables and exponents).
• Factorize common factors from terms.
• Reduce fractions by finding the greatest common divisor.
• Simplify powers and roots using exponent rules.

In our exercise, when simplifying \( g(-2) = 2^{1-(-2)} \), we combined the terms inside the exponent (\(1-(-2)\)) to get \(2^3\), and finally calculated the power, finding that \(2^3 = 8\). Practicing simplifying expressions regularly will make more complex calculations easier to handle.

Powers of 2 are repeatedly multiplying 2 by itself. Powers of 2 show up in binary numbers and computing. Here’s a simple list to illustrate:

• \( 2^0 = 1 \)
• \( 2^1 = 2 \)
• \( 2^2 = 4 \)
• \( 2^3 = 8 \)
• \( 2^4 = 16 \)
• \( 2^5 = 32 \)

Understanding these powers helps in evaluating exponential functions quickly. For instance, when we determined that \( g(-2) = 2^3 \), we recognized that \( 2^3 = 8 \). This fundamental knowledge of powers of 2 is essential for solving many exponential problems in mathematics and computer science.