Problem 5 - Algebra and Trigonometry

Question

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ f(x)=4^{x} ; \quad f(0.5), f(\sqrt{2}), f(-\pi), f\left(\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

\( f(0.5) \approx 2.000; f(\sqrt{2}) \approx 6.350; f(-\pi) \approx 0.015; f\left(\frac{1}{3}\right) \approx 1.587 \).

Step 1: Evaluate f(0.5)

First, substitute \( x = 0.5 \) into the function. We have \( f(0.5) = 4^{0.5} \). Calculate this using a calculator to get \( f(0.5) \approx 2.000 \).

Step 2: Evaluate f(√2)

Substitute \( x = \sqrt{2} \) into the function. We have \( f(\sqrt{2}) = 4^{\sqrt{2}} \). Use a calculator to evaluate this and find \( f(\sqrt{2}) \approx 6.350 \).

Step 3: Evaluate f(-π)

Substitute \( x = -\pi \) into the function. We have \( f(-\pi) = 4^{-\pi} \). Use a calculator to compute this value, which results in \( f(-\pi) \approx 0.015 \).

Step 4: Evaluate f(1/3)

Substitute \( x = \frac{1}{3} \) into the function. We have \( f\left(\frac{1}{3}\right) = 4^{\frac{1}{3}} \). Use a calculator to evaluate this and get \( f\left(\frac{1}{3}\right) \approx 1.587 \).

Key Concepts

Calculator Use in Mathematics

Calculators are invaluable tools in mathematics, especially when working with complex functions, such as exponential functions. An exponential function involves raising a constant to a variable exponent, like what we see in the function \( f(x) = 4^{x} \). Calculating these values manually can be cumbersome and prone to errors due to the precision needed. Calculators do this efficiently and accurately. They are equipped with functions to handle powers and roots, making them perfect for evaluating expressions like those in the exercise.

When using a calculator to evaluate a function, ensure your calculator is in the correct mode (usually degree or radian, based on your problem's requirement). For exponential functions, you typically use the power operation. Enter the base followed by the exponent. It’s important to note that scientific calculators allow for direct input methods, whereas graphical calculators can also plot the function, providing a visual context. Always re-check the mode and entered formula, ensuring there's no input mistake that could lead to incorrect evaluation results.

Rounding Decimals

Rounding decimals correctly is crucial in mathematics to provide results that are both manageable and accurate. The exercise requires rounding to three decimal places, meaning you look at the fourth decimal figure to decide whether to round the third up or keep it the same.

Here’s how rounding works:

• If the digit after your rounding point is 5 or higher, round up.
• If it’s lower than 5, retain the digit.

For example, consider the calculated value \( f(0.5) = 2.000 \, 3 \): The fourth decimal is 3, which is less than 5, so we round down, keeping the value at 2.000. Proper rounding ensures your answers are concise without losing important precision, especially important in fields requiring strict accuracy, like engineering or chemistry.

Practice rounding with different decimal places to become comfortable with the process, ensuring your answers reflect the required level of precision in varied contexts.

Evaluating Functions

Evaluating functions involves substituting a specific value for the variable and calculating the result. With exponential functions like \( f(x) = 4^{x} \), this means you'll substitute each provided value of \( x \) into the exponent and solve.

Here's a simple step-by-step process:

• Identify the function type (here, it's exponential).
• Substitute the given value into the function (e.g., replace \( x \) with \( 0.5, \sqrt{2}, -\pi, \frac{1}{3} \)).
• Use a calculator to get the resulting value.

For instance, when evaluating \( f(\sqrt{2}) \), plug \( \sqrt{2} \) into the function: \( 4^{\sqrt{2}} \). Use your calculator to solve accurately, giving you \( f(\sqrt{2}) \approx 6.350 \).

Understanding the role of each part of the function helps in evaluating it correctly. The base, 4, remains consistent, while the exponent changes with each \( x \) value given, transforming the function's output each time. Regular practice with substitutions and calculations strengthens your function evaluation skills, crucial for tackling more complex mathematical challenges.