Given the following statements: I. The function f(x)=x2sin2x is an odd function. II. The function f(x)=e−xsin⁡x is a neither function. III. The function f(x)=cos⁡(5x)sin2x is an even function IV. The function f(x)=(x−5)e2x is an even function.

Question

Given the following statements:   I. The function f(x)=x2sin2x is an odd function.   II. The function f(x)=e−xsin⁡x is a neither function.   III. The function f(x)=cos⁡(5x)sin2x is an even function   IV. The function f(x)=(x−5)e2x is an even function.

puhnut1m · Accepted Answer

Step 1  we know that: put x=−x and then,  f(−x)=f(x)→ even function  f(−x)=−f(x) odd function  f(−x)≠−f(x)orf(x)→ neither function  Step 2  I) The function f(x)x2sin2x is odd function  So,  put x=−x in function:  f(−x)=(−x)2(sin⁡(−x))2  =x2(−sin)2  f(−x)=x2(sin)2  So, f(−x)−f(x) [this is a even function not odd function] (statement false)

amarantha41 · Accepted Answer

II) The function f(x)=e−xsin⁡x is a neither function:  put x=−x  f(−x)=e−(−x)⋅sin⁡(−x)=ex(−sin⁡x)  f(−x)=−exsin⁡x  ≠−f(x)K or f(x)  So,  This is a neither function (statement true)

karton · Accepted Answer

III) Step 3 The function f(x)=Gx(5x)⋅sin2⁡x is even function So, put x=−x f(−x)=Gx(5(−x))⋅(sin⁡(−x))2 =Gx(−5x)⋅(−sin⁡x)2 f(−x)=Gx(5x)⋅sin2⁡x =f(x) So, This is a even function IV) The function f(x)=(x−5)e2x is an even function. put x=-x So, f(−x)=(−x−5)(e)2(−x) =−(x+5)e−2x ≠ -f(x) or p(x)  So, This is a neither function

Given the following statements: I. The function f(x) = x^{2}

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