Consider, f(x)=(|x-4|/|x|+1). If sum of all distinct possible values of sin -1( sin [f(x)]) is a π+b then find the absolute value of (a+b). [Note: [z] denotes greatest integer function less than or equal to z.]

Question

Consider, f(x)=(|x-4|/|x|+1). If sum of all distinct possible values of sin -1( sin [f(x)]) is a π+b then find the absolute value of (a+b). [Note: [z] denotes greatest integer function less than or equal to z.] (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Range of f ( x ) is [0,4] ∴ sin -1( sin [ f ( x )])= sin -1( sin 0)+ sin -1( sin 1)+ sin -1( sin 2)+ sin -1( sin 3)+ sin -1( sin 4) =0+1+π-2+π-3+π-4=3 π-8 ≡ a π+ b ∴ ( a + b )=|3-8|=5

Q. Consider, $f (x) = \frac{∣ x - 4∣}{∣ x ∣ + 1}$ . If sum of all distinct possible values of $sin^{- 1} (sin [f (x)])$ is $aπ + b$ then find the absolute value of $(a + b)$ .
[Note: [z] denotes greatest integer function less than or equal to $z$ .]

Range of $f (x)$ is $[0, 4]$
$∴ sin^{- 1} (sin [f (x)]) = sin^{- 1} (sin 0) + sin^{- 1} (sin 1) + sin^{- 1} (sin 2) + sin^{- 1} (sin 3) + sin^{- 1} (sin 4)$
$= 0 + 1 + π - 2 + π - 3 + π - 4 = 3 π - 8 \equiv aπ + b$
$∴ (a + b) = ∣3 - 8∣ = 5$

Q. Consider, f(x)=∣x∣+1∣x−4∣​. If sum of all distinct possible values of sin−1(sin[f(x)]) is aπ+b then find the absolute value of (a+b). [Note: [z] denotes greatest integer function less than or equal to z.]

Range of f(x) is [0,4] ∴sin−1(sin[f(x)])=sin−1(sin0)+sin−1(sin1)+sin−1(sin2)+sin−1(sin3)+sin−1(sin4) =0+1+π−2+π−3+π−4=3π−8≡aπ+b ∴(a+b)=∣3−8∣=5

Q. Consider, $f (x) = \frac{∣ x - 4∣}{∣ x ∣ + 1}$ . If sum of all distinct possible values of $sin^{- 1} (sin [f (x)])$ is $aπ + b$ then find the absolute value of $(a + b)$ .
[Note: [z] denotes greatest integer function less than or equal to $z$ .]

Range of $f (x)$ is $[0, 4]$
$∴ sin^{- 1} (sin [f (x)]) = sin^{- 1} (sin 0) + sin^{- 1} (sin 1) + sin^{- 1} (sin 2) + sin^{- 1} (sin 3) + sin^{- 1} (sin 4)$
$= 0 + 1 + π - 2 + π - 3 + π - 4 = 3 π - 8 \equiv aπ + b$
$∴ (a + b) = ∣3 - 8∣ = 5$