Problem 36 Make a hand-drawn graph. Be sure... [FREE SOLUTION]

Chapter 4: Problem 36

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the $x$ -intercepts and the $y$ -intercepts. Check your work using $a$ graphing calculator. $$f(x)=-\frac{6}{x}$$

Short Answer

Expert verified

Domain: $ (-\infty, 0) \cup (0, +\infty) $.No intercepts.Asymptote: $ x = 0 $.

Step by step solution

- Identify the Asymptotes

For the function $ f(x) = -\frac{6}{x} $, identify the vertical and horizontal asymptotes. The vertical asymptote occurs where the denominator is zero, which is at $ x = 0 $. There is no horizontal asymptote for rational functions of this form, but we should consider the behavior as $ x $ approaches $ \infty $ and $ -\infty $.

- Determine the Domain

The domain of $ f(x) = -\frac{6}{x} $ is all real numbers except $ x = 0 $, where the function is undefined. So, the domain is: $ (-\infty, 0) \cup (0, +\infty) $.

- Find the Intercepts

To find the $ x $-intercepts, set $ f(x) = 0 $: \[ -\frac{6}{x} = 0 \] Since there is no value of $ x $ which makes this true, there are no $ x $-intercepts. To find the $ y $-intercepts, set $ x = 0 $. However, since $ x = 0 $ is not in the domain, there are no $ y $-intercepts either.

- Graph the Function

Graph the function with the identified asymptotes and characteristics. Since $ f(x) = -\frac{6}{x} $ is a hyperbola, draw the graph approach the vertical asymptote at $ x = 0 $ and showing that $ f(x) \to 0 $ as $ x \to \infty $ or $ x \to -\infty $.

- Verify with Graphing Calculator

Use a graphing calculator to check the graph and confirm the location of the asymptotes and the shape of the hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain of a Function

The domain of a function represents all the possible input values (usually termed as 'x') that the function can accept without becoming undefined. For the function $ f(x) = -\frac{6}{x} $, the denominator cannot be zero because division by zero is undefined. This means that at $ x = 0 $, the function is not defined. Therefore, the domain of $ f(x) = -\frac{6}{x} $ is all real numbers except zero, which can be written mathematically as: $ (-\infty, 0) \cup (0, +\infty) $.

In this context, remember:

Functions often have domains restricted by denominators
Always check for values that cause the function to be undefined

x-intercepts and y-intercepts

Intercepts are the points where the graph of a function crosses the x-axis or y-axis. To find the x-intercepts, set the function equal to zero and solve for x. For $ f(x) = -\frac{6}{x} $: \[ -\frac{6}{x} = 0 \]

This can never be true since $ -\frac{6}{x} $ can only approach zero but not equal zero. Therefore, there are no x-intercepts for this function.

Now, to find the y-intercepts, set $ x = 0 $. However, since x = 0 is not in the domain of $ f(x) = -\frac{6}{x} $, there are no y-intercepts either.

x-intercepts occur where the function crosses the x-axis
y-intercepts occur where the function crosses the y-axis
Always check if x = 0 is within the domain when looking for y-intercepts

Graphing Rational Functions

When graphing rational functions like $ f(x) = -\frac{6}{x} $, it's important to identify key characteristics such as asymptotes and intercepts. Here is how to graph the function:

Identify vertical and horizontal asymptotes. For $ f(x) = -\frac{6}{x} $, the vertical asymptote is at $ x = 0 $ because the function is undefined there. Horizontal asymptotes describe the function's behavior as $ x \to \infty $ or $ x \to -\infty $.
Evaluate the function at critical points and plot a few points to see the general shape.
Draw the graph approaching the asymptotic lines without crossing them. For $ f(x) = -\frac{6}{x} $, this is a hyperbola that shifts towards the vertical asymptote at $ x = 0 $.

Using a graphing calculator can help verify the graph you have drawn.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as the input values approach certain critical points, typically involving infinity or undefined points. For $ f(x) = -\frac{6}{x} $, the key considerations are vertical and horizontal asymptotes:

The vertical asymptote occurs where the function is undefined. For example, $ x = 0 $
While there may not be a traditional horizontal asymptote, it's still valuable to consider the behavior as $ x \to \infty $ or $ x \to -\infty $. For this function, as x increases, $ f(x) \to 0 $.

These guiding principles are important:

Vertical asymptotes indicate unbounded behavior near specific x-values.
Horizontal or other asymptotes describe how the function behaves at extreme values of x.

91影视

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the \(x\) -intercepts and the \(y\) -intercepts. Check your work using \(a\) graphing calculator. $$f(x)=-\frac{6}{x}$$

Short Answer

Step by step solution

- Identify the Asymptotes

- Determine the Domain

- Find the Intercepts

- Graph the Function

- Verify with Graphing Calculator

Key Concepts

Domain of a Function

x-intercepts and y-intercepts

Graphing Rational Functions

Asymptotic Behavior

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