Problem 9 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.05^{x}\) (a) \(g(-2)\) (b) \(g(120)\) (c) \(g(12)\) (d) \(g(5.5)\)

Step-by-Step Solution

Answer

The solutions are \(g(-2) = 0.907\), \(g(120) = 313089.178\), \(g(12) = 1.795\), and \(g(5.5) = 1.309\).

Step 1: Compute \(g(-2)\)

Substitute \(x = -2\) into the function \(g(x)\): \(g(-2) = 1.05^{-2}\). Use the power rule and a calculator to get the decimal answer, rounding to three decimal places.

Step 2: Compute \(g(120)\)

Substitute \(x = 120\) into the function \(g(x)\): \(g(120) = 1.05^{120}\). Use the power rule and a calculator to get the decimal answer, rounding to three decimal places.

Step 3: Compute \(g(12)\)

Substitute \(x = 12\) into the function \(g(x)\): \(g(12) = 1.05^{12}\). Use the power rule and a calculator to get the decimal answer, rounding to three decimal places.

Step 4: Compute \(g(5.5)\)

Substitute \(x = 5.5\) into the function \(g(x)\): \(g(5.5) = 1.05^{5.5}\). Use the power rule and a calculator to get the decimal answer, rounding to three decimal places.

Key Concepts

Function Evaluation

Function evaluation involves finding the output of a function for a particular input value. Here's what you can do when evaluating the function \( g(x) = 1.05^x \):

• Substitute the given value of \( x \) into the function.
• Perform the necessary calculations to find the result.

For example, if you're given \( g(-2) \), substitute \( -2 \) for \( x \) in the function, giving you \( g(-2) = 1.05^{-2} \). You'll need to calculate \( 1.05^{-2} \) using tools like a scientific calculator to make it easier, especially for non-integer and larger values of \( x \). By understanding the nature of exponential functions like this, you can apply similar steps to any value of \( x \).

Graphing Utility

A graphing utility is a digital tool, such as software on a calculator or a smartphone app, that helps visualize mathematical expressions and compute results efficiently. Here's why it's useful when dealing with exponential functions:

• Allows for easy computation of complex operations like exponentiation, which can be challenging to do manually.
• Helps in verifying your manual calculations by plotting the graph and checking the output results.

In the case of \( g(x) = 1.05^x \), when compute \( g(120) \), it's especially helpful as calculating something like \( 1.05^{120} \) by hand is impractical. Graphing utilities provide a visual check to confirm whether the results calculated manually or with a basic calculator are accurate.

Power Rule

The power rule in mathematics simplifies the computation of exponents within exponential functions. Here's how it applies to functions like \( g(x) = 1.05^x \):

• When using a graphing calculator, enter the base and exponent directly to calculate quickly.
• The power rule states that to calculate \( a^b \), where \( a \) is a constant and \( b \) an exponent, you can use computational tools to streamline finding the solution, especially for fractions and negative exponents.

For example, when evaluating \( g(5.5) = 1.05^{5.5} \), calculators use the power rule internally to deliver correct results to several decimal places. Grasping this concept simplifies your ability to solve exponential problems across different contexts.

Rounding Decimals

Rounding decimals is an essential skill, especially when dealing with the results of exponential functions. Here are some guidelines to consider:

• When instructed to round to a certain number of decimal places, identify the place to round to—three decimal places in this case.
• Look at the digit immediately following this place; if it's 5 or greater, round up. If it's less than 5, keep the digit as is.

For instance, when you've calculated \( g(12) = 1.05^{12} \), and found a result like 1.795856, you round it to 1.796 for three decimal places. By practicing rounding, you ensure that the solutions presented are not only accurate but also align with standard mathematical practices.