Problem 24 - Algebra 2 and Trigonometry

Question

In \(23-34,\) evaluate each function for the given value. Be sure to show your work. $$ \mathrm{f}(x)=x+x^{-5} ; \mathrm{f}(3) $$

Step-by-Step Solution

Answer

\( f(3) = \frac{730}{243} \)

Step 1: Understand the Function

We are given the function \( f(x) = x + x^{-5} \). This means that for any value of \( x \), the function is calculated by adding \( x \) and the inverse fifth power of \( x \).

Step 2: Substitute the Given Value

We need to evaluate the function at \( x = 3 \). Substitute \( x = 3 \) into the function: \( f(3) = 3 + 3^{-5} \).

Step 3: Calculate the Inverse Fifth Power

Calculate \( 3^{-5} \). This means \( \frac{1}{3^5} \). Compute \( 3^5 = 243 \) which means \( 3^{-5} = \frac{1}{243} \).

Step 4: Perform the Addition

Now add \( 3 \) and \( \frac{1}{243} \) together: \( f(3) = 3 + \frac{1}{243} = \frac{729}{243} + \frac{1}{243} = \frac{730}{243} \).

Step 5: Simplify the Fraction (Optional)

Check if the fraction \( \frac{730}{243} \) can be simplified. Since 730 and 243 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding Inverse Powers

When working with inverse powers, the concept might seem daunting at first glance, but it’s not as complicated as it seems.
When you see something like \( x^{-n} \), it means you are dealing with the inverse power of \( x \).

• The negative sign in the exponent indicates that you need to take the reciprocal.
• For instance, \( x^{-5} \) translates to \( \frac{1}{x^5} \).

Let's look at an example using a specific number. If \( x = 3 \), then:

• The inverse fifth power of 3 is \( 3^{-5} = \frac{1}{3^5} \).
• Next, compute \( 3^5 \), which is \( 243 \), resulting in \( 3^{-5} = \frac{1}{243} \).

Understanding inverse powers is crucial, especially when dealing with function calculations where powers of a number need to be inverted for a solution. It’s simply about flipping the value over and calculating the result.

Substitution in Functions

Substitution in functions is a key skill that allows you to evaluate the function at specific points.
Here’s how you can master it:

• Identify the function. In this case, it is \( f(x) = x + x^{-5} \).
• Determine the value at which you need to evaluate the function. For this problem, it's \( x = 3 \).
• Simply replace every occurrence of \( x \) in the function with this value.

For example:

• Substitute \( x = 3 \) into the function: \( f(3) = 3 + 3^{-5} \).
• It's like replacing variables in a formula with specific numbers to find out what the outcome is.

Substitution helps to pinpoint what the function will yield if a specific input is given. It’s a critical step in evaluating and understanding how functions behave at different points.

Mastering Fraction Simplification

Simplifying fractions is all about making them as "small" or as straightforward as possible.
This involves reducing the fraction to its simplest form:

• First, express the fraction clearly. In the problem, we find \( f(3) = 3 + \frac{1}{243} \) to be \( \frac{730}{243} \).
• Then, check the numerator and denominator for common factors.
• If there are any, divide them out. Otherwise, the fraction is already simplified.

Looking at \( \frac{730}{243} \):

• We attempt to find the greatest common factor (GCF) for 730 and 243.
• Since there are no common factors other than 1, the fraction cannot be simplified further.

Fraction simplification is essential to ensure precision and clarity in solutions, helping to yield accurate, concise results in calculations. It’s a crucial skill to have particularly in complex math tasks that involve fractions.