Problem 34 - College Algebra Essentials

Question

Begin by graphing \(f(x)=2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$g(x)=\frac{1}{2} \cdot 2^{x}$$

Step-by-Step Solution

Answer

The graph of function \(g(x) = 0.5 \cdot 2^{x}\) is a vertically compressed version of the graph of function \(f(x) = 2^{x}\), with each y-value halved. The domain of \(g(x)\) is all real numbers, or (-∞, ∞), and its range is \(y >0\), or (0,∞). There is a horizontal asymptote at y=0 for both functions.

Step 1: Graph Base Function

Start by graphing the base function \(f(x) =2^{x}\). This is an exponential growth function, where the output doubles for every increase of 1 in the input value x. The domain is all real numbers and the range is \(y>0\). The graph has a horizontal asymptote at \(y=0\). It passes through the point (0,1) and increases as x increases and approaches 0 as x approaches negative infinity.

Step 2: Apply Transformations

The given function, \(g(x)=\frac{1}{2} \cdot 2^{x}\), is a vertical transformation of the base function \(f(x) = 2^{x}\) by a factor of 0.5. It means that each y-coordinate of the original function is halved. The graph has also a horizontal asymptote at \(y=0\), and it similarly passes through the point (0,0.5). The domain and range remain the same as for the base function.

Step 3: Determine the Domain and Range

For exponential functions, the domain is all real numbers. Therefore, the domain of \(g(x) = 0.5 \cdot 2^{x}\) is also all real numbers, or in interval notation, (-∞, ∞). As the outputs of our function are halved equivalents of \(f(x) = 2^{x}\) and \(f(x)\) has a range of \(y>0\), \(g(x)\)'s range will also be \(y>0\). In interval notation, this is (0, ∞).

Step 4: Confirm with Graphing Utility

Confirm the conclusions by using a graphing tool, such as Desmos, GeoGebra, or a graphing calculator. Plot both \(f(x)\) and \(g(x)\) and observe that \(g(x)\) is indeed a vertically shrunken version of \(f(x)\), with the same general shape, domain, range, and asymptotes.

Key Concepts

Graphing Transformations

Graphing transformations involve shifting or altering a graph in some way. In the provided exercise, we begin with the base function of an exponential graph, \(f(x) = 2^x\). This is a classic example of exponential growth, where each increase in \(x\) results in the output doubling.
When creating the graph for \(g(x)=\frac{1}{2} \cdot 2^x\), we apply a vertical transformation. This transformation involves shrinking the graph vertically by a factor of 0.5. In simpler terms, every point on the graph of \(f(x)\) is moved halfway closer to the x-axis for \(g(x)\).
This type of transformation doesn't affect the horizontal asymptotes or the domain, but it does alter the range as it impacts the y-values of the graph. Transformations like this are especially useful for understanding how different factors in an equation influence the way a graph behaves.

Domain and Range

In mathematical terms, the domain refers to all possible x-values or inputs that a function can accept, while the range comprises all possible y-values or outputs that a function can produce.
For the exponential function \(f(x) = 2^x\), and the transformed function \(g(x) = \frac{1}{2} \cdot 2^x\), the domain is all real numbers. This means you can plug any real number into the function and get a valid output. Therefore, the domain is expressed in interval notation as \((-\infty, \infty)\).
The range for \(f(x)\) is \(y > 0\) because an exponential function never touches the x-axis and will only produce positive values. The transformation doesn’t change the fact that the outputs are still positive, thus the range for \(g(x)\) remains \(y > 0\), written in interval notation as \((0, \infty)\).
Understanding the domain and range is essential as it helps us know the extent of the x-values we can input into the function and the y-values we can get out.

Horizontal Asymptotes

A horizontal asymptote for a function is a horizontal line that the graph of the function approaches as \(x\) goes to infinity or negative infinity. These lines are crucial in understanding the behavior of functions at extreme x-values.
For both \(f(x) = 2^x\) and \(g(x) = \frac{1}{2} \cdot 2^x\), the horizontal asymptote is the line \(y = 0\). This indicates that as the value of \(x\) becomes increasingly negative, the graph of the function approaches the x-axis but never actually touches it.
Essentially, the horizontal asymptote provides a boundary for the y-values of the function when \(x\) is extremely large or small. This boundary does not shift even when a vertical transformation is applied, as seen with \(g(x)\), since it's determined by the base behavior of the exponential function.

Exponential Growth

Exponential growth occurs when the rate of increase in a function is proportional to the current value. This is a hallmark feature of the function \(f(x) = 2^x\), which serves as the base function in the exercise.
The doubling nature of \(f(x) = 2^x\) means that for each unit increase in \(x\), the function's value multiplies by 2, leading to a sharp upward curve on a graph.
When we look at \(g(x) = \frac{1}{2} \cdot 2^x\), the same exponential growth pattern holds since the base of the exponential, which is 2, remains unchanged. However, the starting point is different due to the factor \(\frac{1}{2}\), resulting in a slower rate of growth from the vertical transformation.
The concept of exponential growth is widely applicable across various disciplines like finance, biology, and physics, making it vital to understand.