Question

In Exercises, evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places. \(g(x)=1.075^{x}\) (a) \(g(1.2)\) (b) \(g(180)\) (c) \(g(60)\) (d) \(g(12.5)\)

Short answer

The values of the function \(g(x)=1.075^{x}\) for the given \(x\) values are approximately \(g(1.2) = 1.085\), \(g(180) = 21262758.917\), \(g(60) = 138.953\), and \(g(12.5)= 1.989\).

Step-by-step solution

Evaluation of Function At x = 1.2

Substitute \(x = 1.2\) into the function \(g(x)\) which gives us \(g(1.2)= 1.075^{1.2}\).

Evaluation of Function At x = 180

Substitute \(x = 180\) into the function which gives us \(g(180)= 1.075^{180}\).

Evaluation of Function At x = 60

Substitute \(x = 60\) into the function which gives us \(g(60)= 1.075^{60}\).

Evaluation of Function At x = 12.5

Substitute \(x = 12.5\) into the function which gives us \(g(12.5)= 1.075^{12.5}\).

Find The Decimal Values

Using a calculator or a software, find the decimal values of these expressions, rounded to 3 decimal places.

Key concepts

Exponential Growth

When a quantity increases at a rate proportional to its current value, it's said to exhibit exponential growth. It's commonly represented by the function f(x) = a^{x}, where a is a constant that defines the rate of growth and x represents time or another continuous independent variable. In our exercise, the base 1.075 suggests that the quantity is growing at a rate of 7.5% per period.

For example, if the function represents the population of a species where the rate of growth per year is 7.5%, then g(1.2) would tell us the population after 1.2 years. It’s essential to understand exponential growth to predict the long-term behavior of populations, investments, and compound interest scenarios.

To see this growth visually, plotting the function on a graph shows a curve that gets steeper over time, indicating how the quantity is speeding up as it increases.

Graphing Utilities

Modern graphing utilities, such as calculators and software programs, are essential tools for visualizing and evaluating complex mathematical functions like exponential growth. They allow for quick computations that would otherwise be cumbersome or impossible by hand.

For instance, evaluating g(180) from our exercise by hand is impractical because it involves raising 1.075 to the 180th power – a daunting task without technology. With a graphing utility, not only can we calculate this number easily, but we can also plot the function g(x)=1.075^{x} to observe how the value of g(x) changes as x increases. By using these tools for plotting, we can identify characteristics such as the function's growth rate, intercepts, and asymptotes. This is an excellent way for students to visualize and fully grasp the concept of exponential growth.

Rounding Decimal Places

The rounding of decimal places is a necessary skill when presenting numerical data. It involves reducing the number of digits right of the decimal point to a specific count, making the number simpler and more digestible. When rounding to three decimal places, we look at the fourth decimal place. If this digit is 5 or more, we increase the third decimal place by one, otherwise, we leave it unchanged.

In our exercise, we round the results of g(x) to three decimal places for accuracy and ease of interpretation. For example, if we calculate g(12.5) and get a result like 3.98765, we would round it to 3.988. It's crucial to round correctly because the right amount of precision can impact the interpretation of data, especially in science, finance, and statistics. Proper rounding ensures clear communication and avoids misleading precision that the original data or context cannot support.