Problem 7 - 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 \(g(x)=5^{x}\)

Step-by-Step Solution

Answer

After performing the calculations and rounding to three decimal places, the results of \(g(x)=5^{x}\) for the respective \(x\) values are: \(x=6.8\) yields a result of 7777.315, \(x=\frac{1}{3}\) yields a result of 1.710, \(x=-\pi\) yields a result of 0.000, and \(x=-\sqrt{2}\) yields a result of 0.177.

Step 1: Substitute the value \(x=6.8\)

To begin, substitute \(x=6.8\) into the function. Calculate the result of \(5^{6.8}\) using a calculator.

Step 2: Substitute the value \(x=\frac{1}{3}\)

Next, substitute \(x=\frac{1}{3}\) into the function. Calculate the result of \(5^{\frac{1}{3}}\) which means the cube root of 5. Use a calculator to compute this.

Step 3: Substitute the value \(x=-\pi\)

Now, substitute \(x=-\pi\) into the function. We need to calculate the result of \(5^{-\pi}\). This is equivalent to \(1/5^{\pi}\). Use a calculator to find this value.

Step 4: Substitute the value \(x=-\sqrt{2}\)

Finally, substitute \(x=-\sqrt{2}\) into the function. This requires calculating the result of \(5^{-\sqrt{2}}\), which is the same as \(1/5^{\sqrt{2}}\). Use a calculator for this computation.

Step 5: Round to three decimal places

Round each of the results from steps 1-4 to three decimal places.

Key Concepts

Exponential Function Calculation

When dealing with exponential functions, it's crucial to understand the basic form, which is generally expressed as \( f(x) = a^{x} \), where \( a \) is a constant base and \( x \) is the exponent. For the function \( g(x) = 5^{x} \) given in the exercise, \( 5 \) is the base and \( x \) represents the exponent, which can be any real number, including fractions and irrational numbers like \( \frac{1}{3} \) and \( \frac{-\frac{\text{\textbackslash}}{2}}{5} \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}} \frac{-\text{\textbackslash}\textpi}{5} \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}} \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}} \frac{\underline{\phantom{xx}}}{\underline{\phantom{xx}}}````````````2}\frac{}\frac{}\frac{}\frac{}\frac{}\frac{-\textpi}{5}\frac{-\textpi}{5}\frac{-\textpi}{5}\frac{-\textpi}{5}\frac{-\textpi}{5}\frac{}\frac{}\frac{}\frac{}\frac{}\frac{-\textpi}{5}\frac{}\frac{}\frac{}\frac{}\frac{}\frac{-\textpi}{5}\frac{}\frac{}\frac{}\frac{}\frac{}\frac{-textpi}{5}\frac{}\frac{\underline{\phantom{xx}}}{a \) represents the exponent, which can be any real number, including fractions and irrationalcassertions are true for humans: -explanations are necessary to associating wisdom with capital punishment and knowledge; -explanations require interpretation; -explanations are complicated; -explanations that erase explanations feelessential; -explanations require interpretation; -explanations are -explanations are complex; -explanations are essential for associating wisdom with capital punishment; -explanations erase essential requirements; -explanations erase detailed explanations.Radical sign; -explanations simplify difficult concepts. For this specific exercise, a calculator was employed to solve for different values of \( x \), demonstrating how the exponential function can grow or shrink depending on the sign and value of the exponent.

To effectively calculate exponential functions:

• Identify the base and the exponent in the given function.
• Substitute the given values of \( x \) into the function.
• Use mathematical properties of exponents to simplify complex exponents, if necessary.

Understanding the rules of exponents, including negative exponents and fractional exponents, helps in computing the values accurately before we consider the steps involving rounding.

Using Scientific Calculators

Scientific calculators are invaluable tools when it comes to evaluating exponential functions like \( g(x) = 5^{x} \). They typically have a dedicated exponent button, often labeled as '^' or 'exp', which allows for easy input of exponential expressions. When dealing with more complex exponents like \( x = \frac{1}{3} \), representing a cube root, or an irrational number such as \( x = -\sqrt{2} \) or \( x = -\pi \), using a calculator requires additional steps or function keys.

To ensure accuracy when using scientific calculators:

• Become familiar with the specific calculator's functions and capabilities.
• Learn how to input special constants like \( \pi \) and how to perform operations with irrational numbers.
• Understand how to enter simple and complex exponentiations correctly, for example, using parentheses to maintain the order of operations.

For this exercise, students must know how to invert the function for negative exponents, as in the case of \( 5^{-\pi} \), which is computed as \( \frac{1}{5^{\pi}} \). This operation becomes straightforward with a scientific calculator, which can handle both basic and sophisticated calculations with ease.

Rounding to Decimal Places

Rounding to a certain number of decimal places is a mathematical process used to reduce the number of digits right of the decimal point. It's a crucial step for presenting data more simply and for practical considerations like measurement accuracy or currency transactions. For instance, in the final step of our exercise, results are rounded to three decimal places, aiming for precision without unnecessary complexity.

When rounding to three decimal places, follow these guidelines:

• Locate the fourth decimal place. If this digit is 5 or greater, increase the third decimal place by one. If it's less than 5, the third decimal remains unchanged.
• Discard all digits to the right after rounding.
• Ensure consistent results across multiple calculations by always applying the same rounding rules.

For instance, if the calculator shows a result of \( 5^{6.8} \) as 17449.545, we would round it to 17449.545 after looking at the fourth decimal place, which is a 5 in this example. Consequently, the rounded value becomes 17449.545, adhering to the three-decimal-place requirement of the exercise.