Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Evaluate \(f(x)\) at the given \(x\). Approximate each result to the nearest hundredth. $$f(x)=x^{-0.71}, \quad x=3.8$$

Step-by-Step Solution

Answer

The approximate value of \(f(3.8)\) is 0.42.

Step 1: Understand the Function and the Given Value

The function given is \(f(x) = x^{-0.71}\), and we need to evaluate this function at \(x = 3.8\). This means we will substitute \(x\) with \(3.8\) in the function.

Step 2: Substitute the Value of x

Replace \(x\) in the function with \(3.8\). The expression becomes \(f(3.8) = 3.8^{-0.71}\).

Step 3: Calculate the Power of 3.8

To calculate \(3.8^{-0.71}\), take the reciprocal of \(3.8\) raised to the power of \(0.71\): \[ f(3.8) = \frac{1}{3.8^{0.71}} \] First, calculate \(3.8^{0.71}\) using a calculator or computational tool. This gives approximately 2.38045.

Step 4: Take the Reciprocal

Now take the reciprocal of the result from Step 3: \[ f(3.8) = \frac{1}{2.38045} \] This results in approximately \(0.42006\).

Step 5: Round the Result

Finally, round the result from Step 4 to the nearest hundredth, which is two decimal places. The approximate value is \(f(3.8) \approx 0.42\).

Key Concepts

Evaluating Functions

Evaluating functions means finding the function's output when given a particular input. It involves substituting the variable in the function with the specified value. For example, with the function \( f(x) = x^{-0.71} \), we evaluate it at \( x = 3.8 \).
To do this, simply replace \( x \) with the value 3.8 in the expression, resulting in \( f(3.8) = 3.8^{-0.71} \). This substitution is crucial for determining the output of functions at specific points, providing practical insights into how the function behaves.
When evaluating functions, remember:

• Substitute the given number step by step into the function.
• Perform the mathematical operations as defined by the function.
• Ensure your calculations are accurate, possibly using a calculator for complex computations.

This method allows you to find the exact or approximate value of the function for that input to make informed conclusions.

Negative Exponents

Negative exponents in mathematical expressions indicate reciprocal action on the base number. For instance, \( 3.8^{-0.71} \) means we need to take the reciprocal of \( 3.8^{0.71} \).
Negative exponents can be understood using this simple principle:

• The expression \( x^{-a} \) is the same as \( \frac{1}{x^a} \)

This changes the way we interpret and calculate expressions with negative exponents.
To handle such expressions, compute the positive exponent first, \( 3.8^{0.71} \) in our case, then find the reciprocal to resolve the negative exponent. Expressions with negative exponents can appear complex, but breaking them down makes them manageable.

Function Approximation

Function approximation is a crucial part of evaluating expressions, especially when the results are not whole numbers. Generally, you strive to round the outcomes of your calculations to a specific place for simplicity and clarity.
In our example, the initial computation result was about \( 0.42006 \). Approximating this result to the nearest hundredth involves rounding the value to two decimal places, giving us \( 0.42 \).

• Always ensure that rounding follows standard mathematical rules—if the next digit is five or greater, round up.
• Approximations are vital in mathematics to simplify complex numbers and make them more practical for use in further calculations.

Although approximations might sacrifice some precision, they provide a reasonable and usable value, especially when exact calculations are impractical.