Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. Value \(x=6.8\) \(x=\frac{1}{3}\) \(x=-\pi\) \(x=-\sqrt{2}\) Function \(h(x)=8.6^{-3 x}\)

Step-by-Step Solution

Answer

The results of the calculations are dependent on your specific calculator. Be sure to input the equations exactly as shown and round to three decimal places.

Step 1: Evaluation for \(x=6.8\)

First, replace \(x\) in the function with 6.8: \(h(6.8) = 8.6^{-3 \times 6.8}\). Now, use your calculator to compute the result and round it to three decimal places.

Step 2: Evaluation for \(x=\frac{1}{3}\)

Second, replace \(x\) in the function with \(\frac{1}{3}\): \(h(\frac{1}{3}) = 8.6^{-3 \times \frac{1}{3}}\). Use your calculator to compute the result and round it to three decimal places.

Step 3: Evaluation for \(x=-\pi\)

Next, replace \(x\) in the function with \(-\pi\): \(h(-\pi) = 8.6^{-3 \times -\pi}\). Use your calculator to compute the result and round it to three decimal places.

Step 4: Evaluation for \(x=-\sqrt{2}\)

Lastly, substitute \(x\) in the function with \(-\sqrt{2}\): \(h(-\sqrt{2}) = 8.6^{-3 \times -\sqrt{2}}\). Use your calculator to compute the result and round it to three decimal places.

Key Concepts

Function Evaluation

Function evaluation is a fundamental skill in mathematics, where we find the output of a function given a specific input. This process is like substituting a given value into an equation and calculating the result.

Let's use the function provided, \(h(x)=8.6^{-3x}\). When we're asked to evaluate this function at \(x=6.8\), we replace every instance of \(x\) with 6.8 and simplify. Using a calculator makes this task much easier, especially with exponents. Remember to follow the order of operations: address exponents before multiplication or division.

Correct substitution is essential. For the function \(h(x)\), we would calculate \(h(6.8)=8.6^{-3 \times 6.8}\) by first multiplying -3 by 6.8, then using the calculator to raise 8.6 to that power. Careful input ensures we get the right answer.

Exponential Functions

Exponential functions, like \(h(x)=8.6^{-3x}\), have the general form \(f(x)=ab^{cx+d}\) where \(a\), \(b\), \(c\), and \(d\) are constants, and \(b\) is the base of the exponential term raised to the power of \(cx+d\).

In exponential functions, the variable \(x\) is in the exponent, which drastically changes the function's growth or decay rate. For example, the function provided decays rapidly as \(x\) increases because the base, 8.6, is raised to a negative exponent. Understanding the nature of exponential functions is crucial because they model many real-world phenomena, such as population growth, radioactive decay, and interest in finance.

Calculator Usage

Calculators are indispensable tools for evaluating complex functions or operations like exponents. When working with exponential functions, ensure your calculator is in the correct mode for the type of number input (e.g., radians or degrees for \(\frac{1}{3}\times \text{π}\)).

For accurate results with functions like \(h(x)=8.6^{-3x}\), enter the exponent part as one unit to avoid calculation errors. Many calculators have a caret key (^) to indicate raising to a power, or a dedicated exponent key. Always refer to your calculator's manual for specific instructions. Using calculators effectively is a skill that, once mastered, saves time and reduces error in calculations.

Rounding Numbers

Rounding numbers is a technique used to simplify numbers by reducing the digits while retaining the number's value as accurately as possible. It's often necessary when a non-terminating or an excessively long decimal would complicate further calculations or when a specific level of precision is needed.

For instance, after using a calculator to compute \(h(x)\), you might get a long decimal. If you need to round to three decimal places, look at the fourth decimal place. If it's 5 or higher, round the third decimal place up; if it's 4 or lower, leave the third decimal place as it is. Understanding when and how to round numbers is vital, to ensure that we report numerical answers with appropriate precision.