Problem 10

Question

Use a graphing utility to graph $f(x)=2 / x$ and the function $g$ in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x-1)$$

Step-by-Step Solution

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Answer

The function $g(x)=f(x-1)$ is a horizontal shift of the function $f(x)=2 / x$ to the right by one unit. That is, each point on $f(x)$ is moved one unit to the right to obtain the corresponding point on $g(x)$.

1Step 1: Understanding the First Function

The graph of $f(x)=2 / x$ is a hyperbola. As $x$ approaches zero from the left, $f(x)$ decreases towards negative infinity. As $x$ approaches zero from the right, $f(x)$ increases towards positive infinity. For positive $x$, $f(x)$ is positive and decreases as $x$ increases. For negative $x$, $f(x)$ is negative and increases as $x$ decreases.

2Step 2: Understanding Function Transformation

Now, we look at the function $g(x) = f(x-1)$. This function is a horizontal shift of the function $f(x)$. The $x-1$ inside the function shifts the graph of $f(x)$ one unit to the right. So $g(x)$ is graphically equivalent to the graph of $f(x)$ shifted one unit to the right.

3Step 3: Graphing the Functions

Use a graphing utility to graph the functions $f(x)$ and $g(x)$ in the same viewing window. On close comparison of the two graphs, it can be seen that the graph of $g(x)$ is the same as that of $f(x)$, but shifted one unit to the right.

4Step 4: Describing the Relationship

The relationship between $f(x)$ and $g(x)$ is that $g(x)$ is a horizontal shift of $f(x)$ to the right by 1 unit. That is, each point on $f(x)$ is moved one unit to the right to obtain the corresponding point on $g(x)$.

Key Concepts

Graphing UtilityHyperbolaHorizontal Shift

Graphing Utility

A graphing utility is a digital tool that helps visualize mathematical functions and data. Many students use it because it can illustrate functions in a form that is easier to understand than just reading equations.

These tools can display graphs of complex equations quickly, which makes them great for checking homework answers.
They allow for the exploration of function transformations, such as shifts, stretches, and reflections.
Graphing utilities can also handle a wide range of functions, including polynomials, rational expressions, inequalities, and more.

When using a graphing utility, it's important to choose the correct viewing window, which dictates the portion of the graph that you can see. For the exercise provided, graphing both the functions $ f(x) = \frac{2}{x} $ and $ g(x) = f(x-1) $ in the same window enables you to examine their properties and transformations side by side.

Hyperbola

A hyperbola is a type of smooth curve lying in a plane, which can be represented by the equation $ ax^2 - by^2 = 1 $ or a similar form. It consists of two separate branches that mirror each other.

For the function $ f(x)=\frac{2}{x} $, the graph is in the shape of a hyperbola.
As $ x $ becomes very large or very small, the function values approach zero, creating horizontal asymptotes.
The hyperbolic shape reflects the property that the product of the quantities is constant due to its equation form, simplifying as $ y = \frac{k}{x} $, where $ k $ is a constant.

Understanding the shape and characteristics of a hyperbola helps in predicting how the graph will transform when various modifications are made to the equation, such as shifts or reflections.

Horizontal Shift

Horizontal shifts are one of the most intuitive transformations you can apply to a function's graph. These shifts will move the graph left or right on the Cartesian plane by changing the $ x $-terms.

For the function $ g(x) = f(x-1) $, the notation $ x-1 $ means that every point on the original function $ f(x) $ moves one unit to the right.
This type of transformation does not change the shape of the graph, only its position relative to the origin.

In the context of our problem, the graph of $ g(x) $ maintains the hyperbolic form of $ f(x) $ but is translated horizontally. This means that where $ f(x) $ existed on the coordinate plane, $ g(x) $ now appears shifted one unit to the right. Recognizing horizontal shifts is crucial for understanding more complex transformations and their effects on the graph's behavior.

Problem 10

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