Do you know its property ? -6

Geometry Level 3

Which pair(s) of functions is/are not identical?

(a) log ⁡ e x 3 \log_ex^3 & 3. log ⁡ e x 3.\log_ex

(b) log ⁡ e e x \log_ee^x & e log ⁡ e x e^{\log_ex}

(c) sin ⁡ ( sin ⁡ − 1 x ) \sin(\sin^{-1}x) & sin ⁡ − 1 ( sin ⁡ x ) \sin^{-1}(\sin x)

(d) sin ⁡ 2 x + cos ⁡ 2 x \sin^2x+\cos^2x & sec ⁡ 2 x − tan ⁡ 2 x \sec^2x-\tan^2x

Note : Two functions are called identical if they have same domain and same range.

You can try more such problems of the set Do you know its property?
(c) only (b),(c),(d) (c), and (d) Each pair has identical functions.

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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3 solutions

Sanjeet Raria
Oct 15, 2014

a . a. is completely identical functions since the domain & range are same.

b . b. The first function admits negative x but the second one doesn't. So they are not identical.

c . c. Domain sets are clearly different.

d . d. In spite of being a trigonometrical identity, sec ⁡ 2 x − tan ⁡ 2 x \sec^2 x- \tan^2 x is not defined at x for which cos ⁡ x = 0 \cos x=0 . Which causes the domain difference, hence not identical.

Another nice problem from you @Sandeep Bhardwaj .

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Yeah , exactly explanation dude. Keep it up :)

Sandeep Bhardwaj - 6 years, 8 months ago
Rindell Mabunga
Oct 16, 2014

a: identical

domain of the first is ( − ∞ , + ∞ ) (-\infty , +\infty ) while the second is ( − ∞ , + ∞ ) (-\infty , +\infty )

b: not identical

domain of the first is ( − ∞ , + ∞ ) (-\infty , +\infty ) while the second is [ 0 , + ∞ ) [0 , +\infty )

c: not identical

domain of the first is [ − 1 , 1 ] [-1,1] while the second is ( − ∞ , + ∞ ) (-\infty , +\infty )

d: not identical

domain of the first is ( − ∞ , + ∞ ) (-\infty , +\infty ) while the second is ( − ∞ , − 1 ) ∪ ( 1 , + ∞ ) (-\infty, -1)\cup(1, +\infty)

b , c , d \boxed{b,c,d}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Parth Tandon
Oct 15, 2014

b) 1=x could be + or - 2=x could be +

c) the two fun. have different range.

d)the domain of the 2parts are different

is my exp. correct??????

1 Helpful 0 Interesting 0 Brilliant 0 Confused

c is not correct Edit For inverses, f ( g ( x ) ) = x f(g(x))=x that means domain as well as range are the same set. So in case of c, both domain, & range are different.

Sanjeet Raria - 6 years, 8 months ago

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He is correct as,

Range of sin ⁡ ( sin ⁡ − 1 x ) \sin { (\sin ^{ -1 }{ x } ) } is [-1,1] but

Range of sin ⁡ − 1 ( sin ⁡ x ) \sin ^{ -1 }{ (\sin { x } ) } is [ − π 2 , π 2 ] \left[ \cfrac { -\pi }{ 2 } ,\cfrac { \pi }{ 2 } \right]

Ayush Verma - 6 years, 8 months ago

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Yes indeed!!

Sanjeet Raria - 6 years, 8 months ago

(b),(c),(d) pairs are not identical bro. anyway, how's my problem ?

Sandeep Bhardwaj - 6 years, 8 months ago

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mind blowing problem;;

prajwal kavad - 6 years, 2 months ago

Good problem.

Niranjan Khanderia - 6 years, 2 months ago

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