Problem 8 - Algebra and Trigonometry

Question

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $$ g(x)=\left(\frac{3}{4}\right)^{2 x} ; \quad g(0.7), g(\sqrt{7} / 2), g(1 / \pi), g\left(\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

g(0.7) ≈ 0.698, g(\sqrt{7}/2) ≈ 0.529, g(1/π) ≈ 0.885, g(2/3) ≈ 0.707.

Step 1: Evaluate at x = 0.7

First, substitute \( x = 0.7 \) into the function \( g(x) = \left(\frac{3}{4}\right)^{2x} \). This gives us \( g(0.7) = \left(\frac{3}{4}\right)^{2 \times 0.7} \). Calculate this using a calculator to find \( g(0.7) \approx 0.698 \) (rounded to three decimals).

Step 2: Simplify sqrt(7)/2

Calculate \( \sqrt{7} \) first, then divide it by 2 to simplify \( \frac{\sqrt{7}}{2} \). \( \sqrt{7} \approx 2.646 \), so \( \frac{\sqrt{7}}{2} \approx 1.323 \). We will use this for further evaluation.

Step 3: Evaluate at x = sqrt(7)/2

Now use the approximate value of \( \/sqrt{7}/2 \approx 1.323 \) from the previous step and substitute into the function: \( g\left(\frac{\sqrt{7}}{2}\right) = \left(\frac{3}{4}\right)^{2 \times 1.323} \). Calculating gives \( g\left(\frac{\sqrt{7}}{2}\right) \approx 0.529 \) (rounded to three decimals).

Step 4: Evaluate at x = 1/π

Substitute \( x = \frac{1}{\pi} \) into the function: \( g\left(\frac{1}{\pi}\right) = \left(\frac{3}{4}\right)^{2 \times \frac{1}{\pi}} \). Use a calculator to calculate \( \pi \approx 3.142 \), so \( \frac{1}{\pi} \approx 0.318 \). Therefore, \( g\left(\frac{1}{\pi}\right) \approx \left(\frac{3}{4}\right)^{0.636} \approx 0.885 \) (rounded to three decimals).

Step 5: Evaluate at x = 2/3

Substitute \( x = \frac{2}{3} \) into the function: \( g\left(\frac{2}{3}\right) = \left(\frac{3}{4}\right)^{2 \times \frac{2}{3}} \). Thus, \( g\left(\frac{2}{3}\right) = \left(\frac{3}{4}\right)^{\frac{4}{3}} \approx 0.707 \) (rounded to three decimals).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function given specific inputs. In the context of the exercise, we are evaluating the function \( g(x) = \left(\frac{3}{4}\right)^{2x} \) for different values of \( x \). This function is an example of an exponential function, which is a type of mathematical function where the variable \( x \) appears in the exponent.To evaluate a function like \( g(x) \), follow these steps:

• Identify the given value of \( x \) that you need to substitute into the function.
• Replace \( x \) in the function \( g(x) \) with the given value to obtain a numerical expression.
• Use a calculator to compute the value of this expression.
• Round the result to the required decimal places, if necessary.

By understanding these steps, you can evaluate any mathematical function, determining what output the function produces for specific inputs.

Rounding Decimals

Rounding decimals is an important skill in mathematics, crucial for simplifying numbers and making them easier to work with. When a number is rounded to a certain decimal place, it simplifies to the nearest value based on specific rules. When rounding decimals:

• First, identify the place to which you need to round (e.g., ones, tenths, hundredths, etc.).
• Look at the digit immediately to the right of the desired decimal place.
• If this digit is 5 or greater, increase the last kept digit by 1.
• If the digit is less than 5, leave the last kept digit unchanged.

In our exercise, the results of function evaluations were rounded to three decimal places. For example, when the calculated result was 0.8851, it was rounded to 0.885.

Mathematical Functions

Mathematical functions are relationships between a set of inputs and a set of permissible outputs, with each input corresponding to exactly one output. In mathematical terms, a function is usually written as \( f(x) \), where \( f \) is the function's name and \( x \) is the input.Exponential functions, like the one in our exercise, are special because they involve variables as exponents. The general form of an exponential function is \( f(x) = a^{bx} \), where \( a \) and \( b \) are constants, and \( x \) is the variable.Here are some properties of exponential functions:

• They grow (or decay) at rates proportional to their current value, leading to rapid increases or decreases.
• The base \( a \) must be a positive number, typically greater than 1 for growth.
• If \( a \) is between 0 and 1, the function models exponential decay.

Understanding the behavior of exponential functions is crucial because they appear in diverse areas such as compounded interest in finance, population growth in biology, and radioactive decay in physics.