Let f(x) = [2x2 + 1] and \(g(x)=\left\{\begin{matrix} 2x-3,\,x<0&\\2x+3, x≥0 &\end{matrix}\right.\)where [t] is the greatest integer ≤ t. Then, in the open interval (–1, 1), the number of points where fog is discontinuous is equal to _______.
Let f(x) = [2x2 + 1] and \(g(x)=\left\{\begin{matrix} 2x-3,\,x<0&\\2x+3, x≥0 &\end{matrix}\right.\)where [t] is the greatest integer ≤ t. Then, in the open interval (–1, 1), the number of points where fog is discontinuous is equal to _______.
Correct Answer: 62
Solution and Explanation
\(g(x)=\left\{\begin{matrix} 2x-3,\,x<0&\\2x+3, x≥0 &\end{matrix}\right.\)
The possible points where fog(x) may be discontinuous are
2(2x – 3)2 ∈ I & x ∈ (–1, 0)
2(2x + 3)2 ∈ I & x ∈ [0, 1)
| x ∈ (–1, 0) | x ∈ [0, 1) |
| 2x – 3 ∈ (–5, –3) | 2x + 3 ∈ [3, 5) |
| 2(2x – 3)2 ∈ (18, 50) | 2(2x + 3)2 ∈ [18, 50) |
| So, no. of points = 31 | It is discontinuous at all points except x = 0 of no. points = 31 |
So, the correct answer is: 62.
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
