Math Problem Statement

onsider the following. See Examples 1 and 2.

f(x) =

1

7

−

1

x + 7

7x

Complete the table. (Round your answers to four decimal places.)

x

−7.1

−7.01

−7.001

−7

−6.999

−6.99

−6.9

f(x)

Solution

It looks like the function you're working with is:

f(x)=17−1x+7xf(x) = \frac{1}{7 - \frac{1}{x} + 7x}

You have provided values for xx, and we need to compute f(x)f(x) for each value. Let me calculate the corresponding values of f(x)f(x) and present them rounded to four decimal places.

Given values of xx:

  • x=−7.1x = -7.1
  • x=−7.01x = -7.01
  • x=−7.001x = -7.001
  • x=−7x = -7
  • x=−6.999x = -6.999
  • x=−6.99x = -6.99
  • x=−6.9x = -6.9

Let me calculate the values of f(x)f(x). numpy as np

Define the function

def f(x): return 1 / (7 - (1 / x) + 7 * x)

Values of x

x_values = [-7.1, -7.01, -7.001, -7, -6.999, -6.99, -6.9]

Calculate f(x) for each value of x

f_values = [f(x) for x in x_values]

Round the results to four decimal places

f_values_rounded = [round(value, 4) for value in f_values]

list(zip(x_values, f_values_rounded))Here are the calculated values of f(x)f(x) rounded to four decimal places for the given values of xx:

\hline x & f(x) \\ \hline -7.1 & -0.0235 \\ -7.01 & -0.0239 \\ -7.001 & -0.0239 \\ -7 & -0.0239 \\ -6.999 & -0.0239 \\ -6.99 & -0.0239 \\ -6.9 & -0.0243 \\ \hline \end{array}$$ Let me know if you need any further clarification or details! Would you like to explore this function further, or do you have another question? --- Here are some related questions for you: 1. How does the value of $$f(x)$$ change as $$x$$ approaches -7 from the left and right? 2. What would happen to the function if we evaluated it at values of $$x$$ near zero? 3. How do the behavior of $$f(x)$$ and the function components (like $$\frac{1}{x}$$ and $$7x$$) interact? 4. How would the function behave for values of $$x$$ that are significantly greater than 7 or less than -7? 5. How could you graph $$f(x)$$ to visualize the changes in its behavior across different ranges of $$x$$? **Tip**: When working with functions involving fractions and multiple terms, always check if certain values of $$x$$ could make the denominator zero, as that might lead to undefined values.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Rational Functions
Limits

Formulas

f(x) = 1 / (7 - (1 / x) + 7x)

Theorems

Limit laws
Behavior of rational functions near specific values of x

Suitable Grade Level

Grades 10-12

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