Problem 5 - 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 \(f(x)=3.4^{x}\)

Step-by-Step Solution

Answer

After calculating, the results are: For \(x=6.8\), \(f(6.8) = 3151.203\); For \(x=\frac{1}{3}\), \(f(\frac{1}{3}) = 1.477\); For \(x=-\pi\), \(f(-\pi) = 0.002\); And for \(x=-\sqrt{2}\), \(f(-\sqrt{2}) = 0.350\) (All approximations are rounded to three decimal places).

Step 1: Evaluate function at \(x=6.8\)

Insert the value into the function: \(f(6.8)=3.4^{6.8}\). With a calculator, the result of \(3.4^{6.8}\) will be approximated to 3151.203. Remember to round the value to three decimal places.

Step 2: Evaluate function at \(x=\frac{1}{3}\)

Now insert the fraction value into the function: \(f(\frac{1}{3})=3.4^{\frac{1}{3}}\). Use a calculator to find the value, which is approximately 1.477.

Step 3: Evaluate function at \(x=-\pi\)

This time, \( x = - \pi\). So the function becomes: \(f(-\pi)=3.4^{-\pi}\). With the calculator, find the value of \(3.4^{-\pi}\). It comes out to be approximately 0.002.

Step 4: Evaluate function at \(x=-\sqrt{2}\)

Lastly, insert \(x=-\sqrt{2}\) into the function: \(f(-\sqrt{2})=3.4^{-\sqrt{2}}\). Use a calculator to find this value, and it will approximate to 0.350.

Key Concepts

Calculator Usage

Using a calculator to evaluate exponential functions can greatly simplify the process. Calculators have the capability to handle arithmetic operations and apply them multiple times, which is particularly useful for exponential calculations. When using a calculator, you should:

• Ensure it is set to the correct mode for exponentiation (usually this involves using a dedicated key for exponents).
• Enter the base number first, which in our case is 3.4.
• Use the exponent key (often labeled as "^" or "exp") to enter the exponent value.
• Press the equals button to obtain the result.

By following these steps carefully, you can accurately find values for any exponential function.

Function Evaluation

Function evaluation involves inserting a given input, like a value of x, into a function to find the corresponding output. In our exercise, the function is an exponential function, which means the variable x will serve as the exponent. Let's consider the process:

• Identify the function: Here, it's given as \(f(x) = 3.4^x \).
• Insert each value of x into the function one at a time.
• Use a calculator to compute the result for each insertion.

Evaluating functions accurately will help predict the behavior of the function for different inputs, especially when dealing with non-integer and complex numbers.

Exponents

Exponents denote the number of times a base is multiplied by itself. In the function \(f(x) = 3.4^x \), x is the exponent. Understanding how exponents work is crucial in evaluating functions effectively.

• Positive exponents indicate repeated multiplication, such as \(3.4^2 = 3.4 \times 3.4 \).
• Fractional exponents represent roots, like \(3.4^{1/3} \) corresponds to the cube root of 3.4.
• Negative exponents indicate division, for example \(3.4^{-1} = \frac{1}{3.4} \).

By mastering these concepts, you will gain confidence in evaluating and interpreting functions involving exponents.

Rounding Decimals

Rounding decimals is crucial for maintaining readability and usability of numbers, especially after a calculator has computed a value. When rounding to three decimal places:

• Look at the fourth decimal place: if it’s 5 or more, round up the third decimal place.
• If the fourth decimal is less than 5, keep the third decimal place as it is.

For example, if the calculator gives us 3151.203487, we should round this to 3151.203. Consistently rounding your results as instructed ensures clarity and precision in mathematical communication.