Inverse Trigonometric FunctionshardFree

Inverse Trigonometric Functions — JEE Maths practice question

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

Question

Two functions

f(x)

and

g(x)

are defined as

f(x)=\log_{3}\left|\dfrac{x-2}{x^{2}-10x+24}\right|

and

g(x)=\sin^{-1}\left(\dfrac{2[x]-3}{15}\right)

, then find the number of even integers for which

(f(x)+g(x))

is defined (

[\cdot]

is GIF).

Solution

Answer: 5

Step 1: For

g(x)

: need

\dfrac{2[x]-3}{15}\in[-1,1]

-15\le 2[x]-3\le 15\;\Rightarrow\;-12\le 2[x]\le 18\;\Rightarrow\;-6\le[x]\le 9

x\in[-6,10)

. Step 2: For

f(x)

: need

\dfrac{x-2}{x^{2}-10x+24}

to be defined and non-zero. Factor:

x^{2}-10x+24=(x-4)(x-6)

. So denominator zero at

x=4,6

. Numerator zero at

x=2

. Excluded points:

x\in\{2,4,6\}

. Step 3: Combined domain:

x\in[-6,10)\setminus\{2,4,6\}

. Step 4: Even integers in

[-6,10)

-6,-4,-2,0,2,4,6,8

. Excluding

\{2,4,6\}

-6,-4,-2,0,8

. Step 5: Count:

5

. Answer:

5

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