Problem 47 - Algebra and Trigonometry

Question

graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ f(x)=3^{x} \text { and } g(x)=3^{-x} $$

Step-by-Step Solution

Answer

Both the functions \(f(x) = 3^x\) and \(g(x) = 3^{-x}\) have a horizontal asymptote at \(y = 0\). These functions can also be confirmed using a graphing utility.

Step 1: Graph of the function \(f(x) = 3^x\)

To graph the first function \(f(x) = 3^x\), start by plotting a few points. For example, when \(x = -1\), \(f(x) = 3^{-1} = 1/3\); when \(x = 0\), \(f(x) = 3^0 = 1\); when \(x = 1\), \(f(x) = 3^1 = 3\). As \(x\) moves to the right (increases), \(f(x)\) gets larger, but as \(x\) moves to the left (decreases), \(f(x)\) approaches zero but never reaches it. This shows that the function has a horizontal asymptote at \(y = 0\).

Step 2: Graph of the function \(g(x) = 3^{-x}\)

Now graph the second function \(g(x) = 3^{-x}\). Like before, start by plotting a few points. For example, when \(x = -1\), \(g(x) = 3^1 = 3\); when \(x = 0\), \(g(x) = 3^0 = 1\); when \(x = 1\), \(g(x) = 3^{-1} = 1/3. As \(x\) moves to the left (decreases), \(g(x)\) increases, but as \(x\) moves to the right (increases), \(g(x)\) approaches zero but never reaches it. This shows that the function has a horizontal asymptote at \(y = 0\).

Step 3: Graphing Utility Confirmation

Confirm these hand-drawn graphs using a graphing utility. The graphs should match the ones drawn by hand, including the asymptote at \(y = 0\).

Key Concepts

Asymptotes of Functions

An asymptote of a function is a line that the graph of the function approaches but never actually reaches, no matter how far the graph is extended. Specifically, when graphing exponential functions like the ones in our exercise, the horizontal asymptote is of particular interest.

For the function f(x) = 3^x, as x becomes increasingly negative, the function value approaches zero. This behavior implies that this function has a horizontal asymptote on the line y = 0. Similarly, the function g(x) = 3^{-x} also approaches zero as x becomes increasingly positive, which means it, too, has a horizontal asymptote at y = 0. Understanding where the asymptotes are located helps us to sketch the general shape of the graph of the functions and to predict their behavior at extreme values of x.

Though horizontal asymptotes are most common with exponential functions, other types of functions may have vertical or oblique (slant) asymptotes, each indicating the unattainable boundary of the function's graph.

Exponential Growth and Decay

Exponential functions represent situations where a quantity grows or decays at a rate proportional to its current value. These situations manifest as exponential growth or decay. In our exercise, the function f(x) = 3^x exemplifies exponential growth: as x increases, the value of f(x) increases rapidly. For each unit increase in x, the value of f(x) multiplies by 3.

Conversely, g(x) = 3^{-x} shows exponential decay. As x increases, the value of g(x) gets closer to zero, but never quite reaches it, suggesting the function is decreasing rapidly. These functions are critically important in fields such as biology, economics, physics, and more because they model real-world phenomena like population growth, radioactive decay, and interest accrual with great accuracy.

Graphing Utility Confirmation

Utilizing a graphing utility is an excellent way to confirm the accuracy of hand-drawn graphs of functions, especially when it comes to complex behaviors such as asymptotes. After plotting by hand, a graphing calculator or software can serve as a form of validation.

For instance, graphs of f(x) = 3^x and g(x) = 3^{-x} can be easily verified using such tools. When using a graphing utility, one should observe that the graphs coincide with the hand-drawn functions: f(x) increasing and approaching infinity as x increases, and g(x) decreasing and approaching zero as x increases. Moreover, both functions should have a clear horizontal asymptote at y = 0, a confirmation that gives confidence in the initial sketches and understanding of exponential behavior.