91Ó°ÊÓ

Substitution is a crucial concept in evaluating functions. It's about replacing the variable in a function with a given value. In the original exercise, we were provided the function \( f(x) = x^5 \) and told to evaluate it at \( x = \frac{1}{2} \). This step involves substituting \( \frac{1}{2} \) for every occurrence of \( x \).
So, \( f(x) \) becomes \( f\bigg(\frac{1}{2}\bigg) \). This transformation is the basis for further calculations. It sets the stage for following steps, like exponentiation and simplification, to arrive at the final answer.

Exponentiation means raising a number to a power. It involves multiplying the base by itself however many times indicated by the exponent. In our step, we had to calculate \( \bigg(\frac{1}{2}\bigg)^5 \).
Let's break this down: The base is \( \frac{1}{2} \), and the exponent is 5. This requires us to multiply \( \frac{1}{2} \) by itself five times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \).
Another way to look at it is by raising both the numerator and the denominator to the power of 5 separately:
\[ \bigg(\frac{1}{2}\bigg)^5 = \frac{1^5}{2^5} \].
Simplifying, we know \( 1^5 = 1 \) and \( 2^5 = 32 \), thus we get \( \frac{1}{32} \).

Simplification is the process of making a mathematical expression simpler or more concise while maintaining its value. In this case, after performing the substitution and exponentiation, we are left with:
\[ f\bigg(\frac{1}{2}\bigg) = \frac{1}{32} \].
This represents our final result in its simplest form. Simplification often involves:

• Combining like terms.
• Reducing fractions.
• Performing basic arithmetic operations.

Here, no further work is needed as \( \frac{1}{32} \) is already in its simplest form. This final result is important because it gives us a clear and direct answer to the original function evaluation problem.