Problem 10 - Precalculus with Limits

Question

Evaluating an Exponential Function In Exercises \(7-12,\) evaluate the function at the indicated value of \(x .\) Round your result to three decimal places. $$Function$$ $$f(x)=\left(\frac{2}{3}\right)^{5 x}$$ $$Value$$ $$x=\frac{3}{10}$$

Step-by-Step Solution

Answer

Given function is approximately 0.544 when \(x\) is \(\frac{3}{10}\)

Step 1: Substitute the given value.

First, substitute the value of \(x\) in the function \(f(x)\). This will give \(f\left(\frac{3}{10}\right) = \left(\frac{2}{3}\right)^{5 \cdot \frac{3}{10}}\)

Step 2: Calculate the exponent.

Next, perform the calculation in the exponent. \(5 \cdot \frac{3}{10} = 1.5\). So the function now becomes \(f\left(\frac{3}{10}\right) = \left(\frac{2}{3}\right)^{1.5}\)

Step 3: Evaluate the function.

Calculate \(\left(\frac{2}{3}\right)^{1.5}\) to find the result. The result is approximately 0.544

Key Concepts

exponential function calculation

Understanding exponential functions is crucial when dealing with various areas of mathematics, including algebra and calculus. An exponential function takes the general form of f(x) = a^{bx}, where a is the base, b is the exponent, and x is the variable. Specifically, when evaluating an exponential function, you need to follow the power rule which involves raising the base to a power that's determined by the product of b and x. This may seem straightforward, but the calculations can get complex, especially with non-integer exponents or fractions.

For our example function f(x)=(2/3)^{5x}, it represents growth or decay based on being a fraction (less than 1) raised to a power. In this case, the function exhibits decay since the base is between 0 and 1. Calculating the exact value of such an exponential expression can be challenging without a calculator, especially when dealing with fractional exponents like 1.5, but knowing the fundamental rule that a^{m/n} is the nth root of a to the power of m can aid understanding even if it's complex to calculate manually.

substitute variable values

In mathematics, substituting variable values is a common practice that lies at the heart of solving equations and evaluating functions. When instructed to evaluate a function for a specific value of x, you simply replace the variable x with the given number or expression. This step is crucial as it tailors the general form of the function to a specific case, enabling further calculation.

For instance, in our exercise, f(x)=(2/3)^{5x}, and you need to evaluate the function at x=3/10. By substituting 3/10 into the function in place of x, the exponent becomes 5 * (3/10), which simplifies to 1.5. Therefore, you would then calculate (2/3)^1.5. Substitution helps to visualize the specific snapshot of the function for a particular value and is an essential skill for understanding more advanced mathematical concepts.

rounding decimal places

Rounding decimal places is a numerical method used to reduce the number of digits right of the decimal while retaining a value close to the original number. This is particularly important when you want to simplify a number for ease of communication or when the situation demands a certain level of precision.

In our context, evaluating the function at x = 3/10 yielded a result which was more complex in its decimal form and thus required rounding. The standard rule for rounding is that if the digit in the place you are rounding to is 5 or greater, you round up the previous digit; otherwise, you leave it as is. For example, when rounding the result of our function to three decimal places, if we have a number like 0.544, it will remain the same as no rounding is needed because the fourth decimal place (if available) is less than 5. This rule ensures that the rounded number is both easy to use and as accurate to the original number within the specified level of precision required for the problem.