Question

In Exercises 11–18, graph the function. State the domain and range. $$ f(x)=\frac{-2}{x-7} $$

Short answer

The graph of function \( f(x) = -\frac{2}{x - 7} \) is a hyperbola reflected over the x-axis and shifted right by 7 units with vertical asymptote at x = 7. The domain is \( D = (-\infty, 7) \cup (7, +\infty) \) and the range is \( R = (-\infty, +\infty) \).

Step-by-step solution

Identify the Function

This function \( f(x) = -\frac{2}{x - 7} \) represents a hyperbola. The graph of 1/x is transformed by being reflected over x-axis due to the negative sign in front of the 2, and being shifted to the right 7 units due to the minus sign in front of the 7 in the denominator.

Find the Domain

The domain of the function f(x) is all the values of x for which the function is defined. The function is not defined when the denominator x – 7 = 0. By setting x – 7 = 0 and solving for x, we find x = 7 is not existent in the domain. So, the domain of the function is \( D = (-\infty, 7) \cup (7, +\infty) \).

Find the Range

The range of the hyperbola function \( f(x) = -\frac{2}{x - 7} \) is all real numbers since, depending on x approaching from the left or from the right of 7, the function output can be positive or negative infinity accordingly. So the range of the function is \( R = (-\infty, +\infty) \).

Graph the Function

To graph, plot a vertical asymptote at x = 7, implying that the graph approaches but never reaches x = 7. Plot a few points either side of x = 7 for additional accuracy. Your graph should be two sections of a hyperbola on either side of the line x = 7, with one section above the x-axis and one below, reflecting the negative sign of the expression and the fact that the range is all real numbers.

Key concepts

Domain and Range of Hyperbolas

Understanding the domain and range for any function, especially hyperbolas, is foundational for graphing and comprehending their behavior.

The domain of a hyperbola, similar to the one presented in the exercise, encompasses all the possible input values 'x' for which the function is defined. In general, the only restriction comes from points where the function cannot be calculated, such as divisions by zero. In the example of the function
\( f(x) = -\frac{2}{x - 7} \), the denominator becomes zero when x is equal to 7, which we avoid by excluding this specific value. Thus, the domain is communicated using interval notation as \( D = (-\infty, 7) \cup (7, +\infty) \).

The range, however, refers to the set of possible output values 'y' that the function can produce. For a hyperbola of this type, there is no 'y' value that cannot be the outcome of this function; hence, the range is all real numbers, denoted as \( R = (-\infty, +\infty) \). It is essential to realize that the values approach infinity in both directions as x gets closer to the vertical asymptote but never actually reaches these extreme values.

Asymptotes of Hyperbolas

The concept of asymptotes is central when discussing hyperbolas and their graphs. Asymptotes are lines that the graph of the function approaches but never actually intersects.

In our function \( f(x) = -\frac{2}{x - 7} \), there is a vertical asymptote at x = 7. This is because as the value of 'x' approaches 7, the function's value grows without bound in either positive or negative direction. Since the value at x = 7 is undefined, the graph will get infinitely close to this line but won't cross it. To graph this, draw a dashed line along x = 7 to represent this boundary.

Besides the vertical asymptote, hyperbolas can also have horizontal asymptotes. However, in the case of this function, because the degree of the numerator is less than the degree of the denominator, there is no horizontal asymptote; instead, the arms of the hyperbola will extend infinitely away from the x-axis. Asymptotes serve as guides for sketching the overall shape of the hyperbola.

Hyperbola Transformations

Hyperbola transformations refer to changes in the graph's position and shape in response to certain modifications applied to its function. These transformations make it easier to graph the hyperbola once we understand how they affect the graph.

In the exercise, we see the function \( f(x) = -\frac{2}{x - 7} \). A few key transformations have been applied:

• The negative sign in front of the 2 causes a reflection over the x-axis, flipping the hyperbola so that what would have been above the x-axis is now below, and vice versa.
• The subtraction by 7 in the denominator shifts the entire graph horizontally to the right by 7 units. This is why our vertical asymptote is at x = 7 rather than at the origin.

When graphing the function, first apply these transformations to the standard \(\frac{1}{x}\) hyperbola graph. By understanding and applying hyperbola transformations systematically, one can quickly sketch an accurate graph of the function.