Select The Correct Answer.Consider Function G.g(x)= 5/x-1 +2What Is The Average Rate Of Change Of Function
Understanding the concept of the average rate of change of a function is fundamental in mathematics, especially in calculus and algebra. It provides insight into how a function behaves between two points, offering a snapshot of its overall trend. In this article, we will explore the function \( G.g(x) = \frac{5}{x-1} + 2 \), analyze how to compute its average rate of change over an interval, and guide you through the process step-by-step to select the correct answer among multiple choices.
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What Is the Average Rate Of Change?
The average rate of change of a function between two points measures how much the function's output changes relative to a change in input over a specific interval. Think of it as the slope of the secant line connecting two points on the graph of the function. Mathematically, for a function \( f(x) \), the average rate of change over the interval \([x1, x2]\) is:
\[
\text{Average Rate of Change} = \frac{f(x2) - f(x1)}{x2 - x1}
\]
This formula essentially measures the average "speed" of the function’s change between two points.
Key Points:
- It provides an overall measure of change, unlike the instantaneous rate of change which is given by the derivative at a single point.
- It is useful for understanding the general trend of the function over an interval.
- The units of the average rate of change are typically units of the output variable per units of the input variable.
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Analyzing the Given Function \( G.g(x) = \frac{5}{x - 1} + 2 \)
Before calculating the average rate of change, it’s essential to understand the structure and behavior of the given function:
- Type of Function: Rational function, with a vertical asymptote at \( x=1 \) because the denominator becomes zero.
- Behavior: As \( x \) approaches 1 from the left, \( G.g(x) \) tends to negative infinity; from the right, it tends to positive infinity.
- Range: All real numbers except possibly the values where the function is undefined or approaches asymptotes.
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Step 1: Choose the Interval \([x1, x2]\)
To find the average rate of change, we need specific values for \( x1 \) and \( x2 \). Typically, these are given in the problem options, or we select meaningful points in the domain, avoiding the asymptote at \( x=1 \).
For example, suppose the interval is \([2, 4]\):
- \( x_1 = 2 \)
- \( x_2 = 4 \)
Alternatively, if options suggest different intervals, choose those accordingly.
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Step 2: Calculate \( G.g(x1) \) and \( G.g(x2) \)
Calculate the function values at the selected points:
\[
G.g(2) = \frac{5}{2 - 1} + 2 = \frac{5}{1} + 2 = 5 + 2 = 7
\]
\[
G.g(4) = \frac{5}{4 - 1} + 2 = \frac{5}{3} + 2 = \frac{5}{3} + \frac{6}{3} = \frac{11}{3} \approx 3.6667
\]
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Step 3: Compute the Average Rate of Change
Using the formula:
\[
\frac{G.g(4) - G.g(2)}{4 - 2} = \frac{\frac{11}{3} - 7}{2}
\]
Express 7 as a fraction with denominator 3:
\[
7 = \frac{21}{3}
\]
Subtract:
\[
\frac{11}{3} - \frac{21}{3} = -\frac{10}{3}
\]
Divide by 2:
\[
\frac{-\frac{10}{3}}{2} = -\frac{10}{3} \times \frac{1}{2} = -\frac{10}{6} = -\frac{5}{3}
\]
Result:
\[
\boxed{\text{Average Rate of Change} = -\frac{5}{3}}
\]
This indicates that, on average, over the interval \([2, 4]\), the function decreases at a rate of \( \frac{5}{3} \) units per unit increase in \( x \).
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Interpreting the Result and Choosing the Correct Answer
Depending on the multiple-choice options provided, the result of \( -\frac{5}{3} \) should guide your selection. Common options might include fractions, decimal approximations, or descriptive terms.
Example options:
- A) \( \frac{5}{3} \)
- B) \( -\frac{5}{3} \)
- C) \( 0 \)
- D) \( 2 \)
In this case, the correct choice would be B) \( -\frac{5}{3} \), reflecting the negative average rate of change over the interval.
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Additional Considerations When Calculating Average Rate of Change
- Interval selection: Always ensure that the interval chosen does not include the function's asymptotes or undefined points.
- Multiple intervals: Sometimes, questions may ask for the average rate of change over different intervals; repeat the process accordingly.
- Graphical interpretation: Plotting the function can provide visual insight into whether the function is increasing or decreasing over the interval, aligning with the calculated rate.
- Units and context: In real-world applications, consider what the rate of change represents in context (e.g., speed, growth rate).
Practice Problems for Better Understanding
- For the function \( G.g(x) = \frac{5}{x - 1} + 2 \), compute the average rate of change between \( x=2 \) and \( x=3 \).
- How does the average rate of change differ if you choose \( x=0.5 \) and \( x=2 \), considering the asymptote at \( x=1 \)?
- Graph \( G.g(x) \) and identify the intervals where the function is increasing or decreasing.
Conclusion
Calculating the average rate of change is a vital skill in understanding the behavior of functions, particularly rational functions like \( G.g(x) = \frac{5}{x-1} + 2 \). By selecting appropriate intervals, calculating the function values at those points, and applying the basic formula, you can determine whether the function is increasing or decreasing over that interval and quantify its average rate of change.
Remember, always consider the function's domain and asymptotes to avoid undefined points, and verify your calculations carefully. Whether for academic assessments or practical applications, mastering this concept enhances your overall mathematical proficiency.
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Summary
- The average rate of change measures how a function changes over an interval.
- For \( G.g(x) = \frac{5}{x-1} + 2 \), choose intervals avoiding asymptotes.
- Calculate the function at endpoints, subtract, and divide by the interval length.
- The result indicates the overall trend: positive for increasing, negative for decreasing.
- Practice with different intervals to solidify understanding.
Remember: Always double-check your calculations and interpret the results within the context of the problem to make accurate conclusions.
