Problem 10 - Precalculus: A Concise Course

Question

Evaluate the function at the indicated value of \(x\). Round your result to three decimal places. $$ \begin{aligned} &\text { Function } \quad \text { Valu }\\\ &f(x)=\left(\frac{2}{3}\right)^{5 x} \quad x=\frac{3}{10} \end{aligned} $$

Step-by-Step Solution

Answer

The value of the function \(f(x)\) at \(x=\frac{3}{10}\) is approximately 0.577.

Step 1: Substitute the value of x into the function

Replace \(x\) by the given value \(\frac{3}{10}\) in the function \(f(x)=(\frac{2}{3})^{5x}\). Thus, it becomes \(f(\frac{3}{10})=(\frac{2}{3})^{5 \times \frac{3}{10}}\).

Step 2: Simplify the exponent

Perform the multiplication in the exponent, \(5 \times \frac{3}{10}=1.5\), to simplify the function as \(f(\frac{3}{10})=(\frac{2}{3})^{1.5}\).

Step 3: Calculate the value of the function

The final step is to calculate the value of \((\frac{2}{3})^{1.5}\). Upon calculating, the approximate value is 0.577.

Key Concepts

Function Evaluation in Exponential Functions

When dealing with functions, function evaluation simply means replacing the variable with a given value and solving the equation. Here, our function is exponential: \( f(x) = \left(\frac{2}{3}\right)^{5x} \). We need to evaluate it at \( x = \frac{3}{10} \). This process involves:

• Substitute the variable \( x \) with \( \frac{3}{10} \) in the function.
• This turns the equation into \( f\left(\frac{3}{10}\right) = \left(\frac{2}{3}\right)^{5 \times \frac{3}{10}} \).

We're essentially finding the value of the function for this specific input value. It’s a straightforward substitution process. Next, we simplify the exponent to continue evaluating it.

Understanding Exponents in Exponential Functions

Exponents in mathematics are a way of expressing how many times a number, known as the base, is multiplied by itself. In our function, we have an exponent presented as \( \left(\frac{2}{3}\right)^{5x} \), where \( 5x \) is the exponent.This part involves multiplying 5 by our substituted value of \( x = \frac{3}{10} \):

• Perform the multiplication: \( 5 \times \frac{3}{10} = 1.5 \).
• Now the exponent is simplified to 1.5, resulting in the expression \( \left(\frac{2}{3}\right)^{1.5} \).

Exponents like 1.5, which are not integers, are perfectly valid and are known as fractional exponents. Solving this involves calculating what is often called the root; here, a mix of a square root and a cube, but simplified using calculators.

Rounding the Result

Rounding means adjusting the decimal point of a number to make it simpler and easier to use or present. In mathematics problems like ours, it's crucial to round numbers to a specific decimal place. Here, the instruction is to round to three decimal places.Once the value of the function is computed, we arrive with a decimal answer. Let's say you find the result of \( \left(\frac{2}{3}\right)^{1.5} \) using a calculator:

• You might get a number like 0.576898.
• Rounding this to three decimal places gives us 0.577.

When rounding, observe the fourth decimal point digit: if it's 5 or more, round up the third digit; if less, keep it the same. Here, the calculation tells us to keep things precise, and for this problem, the rounded result is 0.577.