Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. \(f(x)=2(5)^{x},\) for \(f(-3)\)

Step-by-Step Solution

Answer

The value of \( f(-3) \) is 0.0160.

Step 1: Substitute the Value

Substitute the given value of \( x = -3 \) into the function \( f(x) = 2(5)^x \). This will give us the expression \( f(-3) = 2(5)^{-3} \).

Step 2: Simplify the Exponent

Calculate \( (5)^{-3} \), which is equivalent to \( \frac{1}{5^3} \). This simplifies to \( \frac{1}{125} \).

Step 3: Multiply by the Coefficient

Multiply the result from Step 2 by the coefficient 2 as given in the function: \( f(-3) = 2 \times \frac{1}{125} \).

Step 4: Calculate the Final Result

Now perform the multiplication: \( 2 \times \frac{1}{125} = \frac{2}{125} = 0.016 \).

Step 5: Verification and Rounding

Verify the calculation and ensure the answer is rounded to four decimal places. In this case, the answer is already \( 0.0160 \).

Key Concepts

Exponential Functions

Exponential functions are an important class of functions in mathematics where the variable appears in the exponent. In general, they take the form \( f(x) = a(b)^x \), where \( a \) is a constant and \( b \) is the base of the exponential. These functions tend to grow very quickly, especially when \( b > 1 \). They can model a variety of real-world situations, such as population growth, radioactive decay, and interest calculations.

Our task involves evaluating an exponential function \( f(x) = 2(5)^x \) with a specific value of \( x = -3 \). To do this, we substitute \( x \) with \(-3\), resulting in \( f(-3) = 2(5)^{-3} \). In exponential functions, the base is raised to the power indicated by the exponent. Here, it is important to understand how to deal with negative exponents, a concept we will discuss soon.

Understanding exponential functions requires knowing how to manipulate exponents and understanding their behavior as the base is positive. They are not just mathematical curiosities; they have powerful applications in diverse fields.

Rounding Decimal Places

When working with numbers, precision is key. Sometimes, though, it's necessary to round numbers to make them easier to work with or to match a required level of accuracy. To round numbers to a specific number of decimal places, you look at the digit immediately to the right of your desired decimal place.

In our exercise, the final calculation resulted in \( 0.016 \). We need to check if it should be rounded to four decimal places. To do so, consider the next digit. If this digit is 5 or higher, we increase the last significant digit; if not, we leave it as it is. For instance, rounding \( 0.016 \) to four decimal places means adding a trailing zero, resulting in \( 0.0160 \).

This practice helps in maintaining consistency, especially in fields like engineering, finance, or when reporting scientific data, where precision matters. Rounding not only makes numbers easier to communicate but also aligns with conventions and reduces unnecessary detail.

Negative Exponents

Negative exponents might be a little confusing initially, but they follow a simple rule. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. Essentially, \((b)^{-n} = \frac{1}{b^n}\). This means you "flip" the base into the denominator and make the exponent positive.

In our function evaluation, \((5)^{-3}\) becomes \( \frac{1}{5^3} \). Calculating further, \( 5^3 = 125 \), so \( (5)^{-3} = \frac{1}{125} \). Understanding this conversion is essential when dealing with exponential functions, especially when they decrease quickly as the exponent becomes more negative.

Grasping how to work with negative exponents opens doors to solving complex equations and understanding growth models that decrease or decay. Whether you're dealing with finance, where depreciation models might be used, or physics, where certain decay processes occur, mastering negative exponents aids in comprehending a wide range of phenomena.