Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, graph the transformation of \(f(x)=2^{x}\). Give the horizontal asymptote, the domain, and the range. \(f(x)=2^{-x}\)

Step-by-Step Solution

Answer

Horizontal asymptote: \(y = 0\); Domain: \((- \infty, \infty)\); Range: \((0, \infty)\).

Step 1: Identify the transformation

The given function is \(f(x) = 2^{-x}\). This is a transformation of the original function \(f(x) = 2^x\). The transformation involves a reflection over the y-axis due to the negative exponent.

Step 2: Graph the original function

To understand the transformation, first graph the original function \(f(x) = 2^x\). This is an exponential function that grows rapidly, has a horizontal asymptote at \(y = 0\), domain of all real numbers \((-\infty, \, \infty)\), and range \((0, \, \infty)\).

Step 3: Apply the horizontal reflection

Reflect the graph of \(f(x) = 2^x\) over the y-axis to get \(f(x) = 2^{-x}\). This means every point \((x, y)\) on \(2^x\) is transformed to \((-x, y)\) on \(2^{-x}\).

Step 4: Graph the transformed function

The graph of \(f(x) = 2^{-x}\) will now start at \(y = 1\) when \(x = 0\), decay towards the horizontal asymptote as \(x\) increases, and will go to infinity as \(x\) decreases. The shape is like a mirror image of \(f(x) = 2^x\).

Step 5: Define the horizontal asymptote

For \(f(x) = 2^{-x}\), the horizontal asymptote remains the same as the original function, at \(y = 0\), because the output doesn't fall below zero.

Step 6: Determine the domain

The domain of \(f(x) = 2^{-x}\) is all real numbers \((-\infty, \infty)\), just as in \(f(x) = 2^x\). The function is defined for every real number \(x\).

Step 7: Determine the range

The range of \(f(x) = 2^{-x}\) is \((0, \infty)\) because, as an exponential function reflected over the y-axis, it never outputs zero or negative values, only positive numbers.

Key Concepts

Function Transformations

Function transformations help us modify the basic shape and position of functions on a graph. In the given exercise, we start with the original function, \(f(x) = 2^x\), and transform it into \(f(x) = 2^{-x}\). This transformation involves a reflection over the y-axis.

A reflection over the y-axis essentially flips the function. For every point \((x, y)\) on the graph of \(2^x\), there is a corresponding point \((-x, y)\) on the graph of \(2^{-x}\).

Here are some essential points to keep in mind regarding function transformations:

• Reflections change the direction in which the graph heads. With \(f(x) = 2^{-x}\), the graph now decays as \(x\) increases, rather than growing.
• Translations can shift the graph up, down, left, or right, though they are not applied in this specific transformation.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches, but never truly reaches. For both the original function \(f(x) = 2^x\) and its transformed version \(f(x) = 2^{-x}\), the horizontal asymptote is \(y = 0\). This is because, in exponential functions, the output approaches zero but never becomes negative or zero.

The presence of a horizontal asymptote informs us about the end behavior of the function. Notably, as \(x\) approaches infinity, \(2^{-x}\) will get very close to zero, but won't actually touch or drop below this line.

Domain and Range

Understanding the domain and range of a function helps us see the possible input and output values.
For both \(f(x) = 2^x\) and \(f(x) = 2^{-x}\), the domain is all real numbers, \((-\infty, \, \infty)\). This means you can input any real number into the function.
The range for \(f(x) = 2^{-x}\) is \((0, \infty)\), indicating that the outputs are always positive numbers. Since exponential functions approach zero but never quite get there, zero isn't part of the range.
In simpler terms, you can think of the domain as where you can go horizontally on the graph, while the range is how high or low (but not below zero) the graph can get vertically.

Graphing Functions

Graphing functions is a key skill for visualizing equations and understanding their behavior. Let's consider the graph of \(f(x) = 2^{-x}\).
1. Start by graphing the basic exponential function \(f(x) = 2^x\).2. Apply the transformation by reflecting it over the y-axis to get \(f(x) = 2^{-x}\).
This results in a graph that starts at \(y = 1\) when \(x = 0\) and then decreases toward the horizontal asymptote \(y = 0\) as \(x\) increases, while increasing steeply as \(x\) becomes more negative.
By following these steps, you not only learn how to plot the function but also how to anticipate its behavior by understanding transformations and asymptotic behavior. Using graphing aids, like plotting software or graphing calculators, can help you further visualize these concepts.