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In mathematics, a function is a relation between a set of inputs and a set of possible outputs. For function evaluation, we substitute the input (known as the variable) with a specific value. In the given exercise, the function is defined as: \[ f(x) = 4^x \] Here, we are asked to find the value of the function when \( x = \frac{1}{2} \). This means substituting \( \frac{1}{2} \) wherever we see \( x \) in the function definition. So, you would plug \( \frac{1}{2} \) into the function: \[ f\left(\frac{1}{2}\right) = 4^{\frac{1}{2}} \] This gives us a new mathematical expression to solve. The process of replacing the variable in a function with a number and solving for the output is what we call function evaluation.

An exponent indicates how many times a number, known as the base, is multiplied by itself. In our given function, the base is 4, and the exponent is \( \frac{1}{2} \). \[ 4^{\frac{1}{2}} \] In general, the expression of \( a^b \) means 'a raised to the power of b'. Understanding how exponents work is essential for simplifying such expressions. When the exponent is a fraction, this usually represents a root. Specifically, the exponent of \( \frac{1}{2} \) indicates the square root. Hence, in this case, we calculate the square root of 4: \[ 4^{\frac{1}{2}} = \sqrt{4} = 2 \] Hence, the result is 2. This is because 2 multiplied by itself (2 x 2) gives 4. Remember that mastering exponents can simplify a lot of mathematical problems, especially those involving roots and powers. Breaking the problem down step-by-step can help clarify complex concepts.

Calculators are incredibly useful tools for solving mathematical problems, especially those involving complex operations like exponents. There are a few steps to follow to accurately use a calculator for function evaluation:

• Ensure the calculator is in the correct mode (usually standard or scientific mode for basic calculations).
• Input the base number first (in this case, 4).
• Use the exponentiation function, often marked as \( \wedge \) or \( ^ \) or sometimes requiring a special keypress combination.
• Enter the exponent (\( \frac{1}{2} \)), which may require a specific format for fractional exponents, like using parentheses or a special fraction key.

After keying in the operation correctly, the calculator will display the result. For \( 4^{\frac{1}{2}} \), pressing the correct keys will typically show 2. To ensure accuracy, check that you have used the correct sequence of operations. It's good practice to verify your result by doing a quick mental approximation or using another method. Always round your final answer to the required decimal place, here to the nearest hundredth, although in this case rounding 2 gives us 2.00.