Problem 32 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=\left(\frac{x^{-1}}{(2 x)^{-2}}\right)^{-1} ; \mathrm{f}(8) $$

Step-by-Step Solution

Answer

\(\mathrm{f}(8) = \frac{1}{32}.\)

Step 1: Substitute the Value of x into the Function

We need to substitute the given value of \(x = 8\) into the function \(\mathrm{f}(x)\). This yields: \[\mathrm{f}(8) = \left(\frac{8^{-1}}{(2 \cdot 8)^{-2}}\right)^{-1}.\]

Step 2: Simplify the Denominator

Calculate \((2 \cdot 8)^{-2}\). Start by simplifying \(2 \cdot 8 = 16\). Then, \(16^{-2} = \frac{1}{16^2} = \frac{1}{256}\). So, the expression becomes: \[\left(\frac{8^{-1}}{\frac{1}{256}}\right)^{-1}.\]

Step 3: Simplify the Numerator

Calculate \(8^{-1}\) as \(\frac{1}{8}\), changing the expression to: \[\left(\frac{\frac{1}{8}}{\frac{1}{256}}\right)^{-1}.\]

Step 4: Simplify the Fraction

To divide the fractions, multiply by the reciprocal of the denominator: \[\frac{1}{8} \div \frac{1}{256} = \frac{1}{8} \times 256.\] This simplifies to \(\frac{256}{8} = 32\). So, we have \[32^{-1}.\]

Step 5: Evaluate the Final Expression

Now, \(32^{-1} = \frac{1}{32}.\) Thus, \(\mathrm{f}(8) = \frac{1}{32}.\)

Key Concepts

Understanding Exponents

Exponents represent repeated multiplication of a number by itself. For example, when we see the expression \(x^3\), it means \(x\) multiplied by itself two more times, i.e., \(x \cdot x \cdot x\). Similarly, \(x^{-1}\) indicates the reciprocal of \(x\), or \(\frac{1}{x}\).

• Positive exponents mean multiplying the base many times.
• Negative exponents imply taking the reciprocal of the base.
• Zero exponent \(x^0\) results in \(1\) for any non-zero \(x\).

When evaluating a function involving exponents, it's crucial to understand these rules. They help in transforming complex expressions into simpler forms. In our solution, negative exponents were converted into reciprocals to simplify calculation, demonstrating their powerful role in function evaluation.

The Art of Function Simplification

Function simplification is the process of making a function easier to understand or calculate by reducing its complexity. It often involves simplifying terms, factoring, and using arithmetic operations like addition, subtraction, multiplication, or division.

• Always perform operations inside parentheses first.
• Simplify exponents early to streamline calculations.
• Identify common factors in expressions to reduce them.

In the given problem, the function \(f(x) = \left(\frac{x^{-1}}{(2x)^{-2}}\right)^{-1}\) was simplified by substituting the given value of \(x\), using arithmetic operations inside the fractions, and applying the reciprocal of each segment to reduce its complexity. These steps illustrate how simplification can be methodically executed for precise results.

Exploring Reciprocals

Reciprocals play a vital role in mathematics, especially when dealing with division and negative exponents. The reciprocal of a number \(a\) is simply \(\frac{1}{a}\). It helps turn division into a multiplication operation, which is easier to handle.

• Finding a reciprocal involves flipping the numerator and denominator.
• Multiplying any number by its reciprocal yields 1.
• Negative exponents inherently represent reciprocals.

For instance, in the solution of the exercise, the reciprocal was used in multiple ways. The negative exponent \(8^{-1}\) was replaced by \(\frac{1}{8}\), and a division-related fraction \(\frac{1}{8} \div \frac{1}{256}\) was turned into a multiplication \(\frac{1}{8} \times 256\). Such manipulations simplify calculations, showcasing how understanding reciprocals can make math more manageable.