FunctionsmediumFree

Functions: Number Solutions Equation Cases Cases

JEE Maths question with a full step-by-step solution.

Question
Number of solutions of the equation f(x−1)+f(x+1)=sin⁡αf(x-1)+f(x+1) = \sin\alpha, 0<α<π20 < \alpha < \dfrac{\pi}{2}, where
f(x)={1−∣x∣;∣x∣≤10;∣x∣>1f(x) = \begin{cases} 1-|x|; & |x| \leq 1 \\ 0; & |x| > 1 \end{cases}
is.
Solution
Answer: 4
Step 1: Compute g(x)=f(x−1)+f(x+1)g(x) = f(x-1)+f(x+1) f(x−1)f(x-1) is a tent centered at x=1x=1, zero outside [0,2][0,2]. f(x+1)f(x+1) is a tent centered at x=−1x=-1, zero outside [−2,0][-2,0].
g(x)={x+2−2≤x≤−1−x−1≤x≤0x0≤x≤12−x1≤x≤20otherwiseg(x) = \begin{cases} x+2 & -2 \leq x \leq -1 \\ -x & -1 \leq x \leq 0 \\ x & 0 \leq x \leq 1 \\ 2-x & 1 \leq x \leq 2 \\ 0 & \text{otherwise} \end{cases}
 gg has two peaks of height 1 at x=−1x = -1 and x=1x = 1. Step 2: Count intersections with y=sin⁡αy = \sin\alpha Since sin⁡α∈(0,1)\sin\alpha \in (0,1), the horizontal line y=sin⁡αy = \sin\alpha intersects each of the four rising/falling segments exactly once:
g(x)=sin⁡α has exactly 4 solutionsg(x) = \sin\alpha \text{ has exactly } 4 \text{ solutions}
 Answer: 4
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Functions · easy
The range of the function y=[x2]−[x]2y=[x^{2}]-[x]^{2}, x∈[0,2]x\in[0,2] (where [⋅][\cdot] denotes the integral part) is:
Functions · easy
If [x][x] denotes the integral part of xx, then domain of the function f(x)=3−x(x−1)(x−2)(x−3)+sin⁡−1 ⁣[3x−22]f(x)=\dfrac{\sqrt{3-x}}{(x-1)(x-2)(x-3)}+\sin^{-1}\!\left[\dfrac{3x-2}{2}\right] is:
Functions · easy
The number of integers in the domain of f(x)=1ln⁡cos⁡−1xf(x) = \dfrac{1}{\sqrt{\ln\cos^{-1}x}}.
Functions · easy
Let [⋅][\cdot] denote the greatest integer function. If the domain of f(x)=cos⁡−1(4x+2[x]3)f(x)=\cos^{-1}\left(\dfrac{4x+2[x]}{3}\right) is [α,β][\alpha,\beta], then 12(α+β)12(\alpha+\beta) is equal to

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath