Problem 48 - College Algebra

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$f(x)=2.7(4)^{-x+1}+1.5, \text { for } f(-2)$$

Step-by-Step Solution

Answer

The evaluated function for \( f(-2) \) is 174.3.

Step 1: Substitute the value of x

The exercise asks us to evaluate the function \( f(x) = 2.7(4)^{-x+1} + 1.5 \) for \( x = -2 \). We replace \( x \) with \( -2 \) in the function.\[f(-2) = 2.7(4)^{-(-2)+1} + 1.5\]

Step 2: Simplify the exponent

Simplify the exponent in the term \( (4)^{-(-2)+1} \). Calculate \( -(-2) + 1 \).\[-(-2) + 1 = 2 + 1 = 3\]

Step 3: Evaluate the base

Now substitute the simplified exponent back into the equation.\[f(-2) = 2.7(4)^3 + 1.5\]Calculate \( (4)^3 \).\[(4)^3 = 64\]

Step 4: Calculate the expression

Substitute the value of \( (4)^3 \) back into the function.\[f(-2) = 2.7 imes 64 + 1.5\]Multiply the terms.\[2.7 imes 64 = 172.8\]Then, add the constant term.\[f(-2) = 172.8 + 1.5 = 174.3\]

Step 5: Round the Result

Check if the result needs to be rounded to four decimal places. Since the number 174.3 already ends at one decimal place, it remains the same: 174.3.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical expression where a constant base is raised to a variable exponent. In our exercise, the function given is \( f(x) = 2.7(4)^{-x+1} + 1.5 \). Here, 4 is the base and \(-x+1\) is the exponent. Exponential functions are known for their rapid growth or decay based on whether the base is greater than or less than one.

• If the base is greater than one, as in our case (base = 4), the function grows as the variable increases.
• When the base is between 0 and 1, the function tends to decay or get smaller as the variable increases.

Exponential functions are crucial in modeling real-world phenomena like populations growth, radioactive decay, and interest calculations.
Being familiar with how to evaluate these functions helps in understanding the dynamics of such processes.

Substitution in Functions

Substituting in functions involves replacing the variable with a given value to find the function's output. In our example, we want to compute \( f(-2) \). To do this, we replace \( x \) with \( -2 \) in the equation: \[ f(-2) = 2.7(4)^{-(-2)+1} + 1.5 \] This substitution is like plugging in a value to see how the function reacts.
It can often involve multiple steps since the initial replacement might lead to more operations.

• First, replace the variable with the given number.
• Second, solve any resulting operations within the exponent or equation.

This technique is foundational for analyzing how functions behave for various inputs, making it applicable across almost all fields of study.

Simplifying Exponentials

Simplifying exponentials is about reducing the powers in exponential expressions for easier computation. In the given problem, once we substitute \( x = -2 \), the expression becomes \( (4)^{-(-2)+1} \). Our job is to simplify the exponent first:\[-(-2) + 1 = 2 + 1 = 3\] Thus, the function transforms to \( (4)^3 \).
Performing these simplifications accurately is essential because it directly affects the outcome of the evaluation.

• Pay attention to the signs in front of numbers; a negative sign can change the direction of simplification.
• Always resolve what's inside the bracket first before moving on.

With practice, simplifying such expressions becomes a quicker and more intuitive process.

Rounding Decimal Places

Rounding decimal places means adjusting a numerical result to a specific degree of precision. In our function evaluation, after the calculations, the result was \( 174.3 \). The exercise specifies rounding to four decimal places, but since our result already has fewer decimals, it stands as \( 174.3 \).
Rounding is frequently used in real life to simplify results, make them more manageable, or meet specific requirements.

• Determine the decimal place you need.
• Check the number directly after this place to decide whether to round up or leave it.
• In our case, as there is no need to round to more decimal places, the number stays as it is.

Being thorough with the rounding process ensures accuracy and precision in both academic and practical scenarios.