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66. \( f(x)=\frac{x^{2}-x+5}{x} \)

Ask by Bolton Stanley. in the United States
Feb 25,2025

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Answer

\( f(x) = x - 1 + \frac{5}{x} \)

Solution

Calculate or simplify the expression \( (x^2-x+5)/x \). Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{\left(x^{2}-x+5\right)}{x}\) - step1: Calculate: \(x-1+\frac{5}{x}\) The simplified form of the function \( f(x) = \frac{x^{2}-x+5}{x} \) is \( f(x) = x - 1 + \frac{5}{x} \).

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The function \( f(x) = \frac{x^{2} - x + 5}{x} \) can be rewritten to show its components more clearly. By simplifying, you can express it as \( f(x) = x - 1 + \frac{5}{x} \). This form makes it easier to analyze its behavior, such as identifying asymptotes and examining limits as \( x \) approaches zero. In real-world applications, rational functions like this one can model various scenarios, such as rates of change, population growth, or resource consumption. For instance, if you consider \( f(x) \) to represent costs over time, understanding how the function changes can help with budgeting strategies or planning investments based on how \( x \) (time or quantity) influences overall costs.

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1.3.2 Make a conjecture with regard to \( r^{n} \) and \( S_{n} \) as \( n \rightarrow \infty \) (2) 1.4 CASE 3: \( r=1 \) 1.4.1 What is the sum of the geometric series \[ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=1 \text { ? } \] 1.4.2 Make a conjecture with regard to \( r^{n} \) and \( S_{n} \) as \( n \rightarrow \infty \) (2) 1.5 CASE 4: \( r=-1 \) 1.5.1 What is the sum of the geometric series \[ S_{n}=a+a r+a r^{2}+\ldots a r^{n-1} \text { if } r=-1 ? \] 1.5.2 Do the sums above approach some finite particular number as \( n \rightarrow \infty \) i.e. is the sequence divergent or convergent? 1.6 CASE 5: \( -1<r<1 \) REQUIREMENTS: - One A4 papers - Provided grid 1.6.1 Write THREE possible values of \( r \) such that \( -1<r<1 \). 1.6.2 Step 1. Cut the A4 size paper along the longest side into two equal Rectangles and define their areas to be 16 unit \( ^{2} \). 1.6.3 Step 2. Place one half of the rectangle in Step 1 on the desktop and cut the other half along the longest side in to two equal rectangles. 1.6.4 Step 3. Place one half of the rectangle in Step 2 on the desktop and cut the other half along the longest side into two equal rectangles. 1.6.5 Step 4. Continue with the procedures from Step 3 until you find it too difficult to fold and cut the piece of paper you are holding. 1.6.6 Step 5. The first piece of paper you placed on the desktop has an area of \( \frac{1}{2} \) the area of the A4. The second piece of paper has an area of \( \frac{1}{4} \) the area of the A4. Write the areas of the next three pieces of paper. 1.6.7 Explain why these areas form a geometric seauence
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