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2.42. Consider f (x) = x3????x and g(x) = x2????1 on x 2 [????1;1]. (a) Verify that f (x) is an odd function and g(x) is an even function, meaning f (????x) = ????f (x)) and g(????x) = g(x). (b) Directly compute that h f (x);g(x)i = 0.

Respuesta :

Cricetus

Answer:

f(x) is an odd function and g(x) is an even function

Step-by-step explanation:

Even Function :

A function f(x) is said to be an even function if

f(-x) = f(x) for every value of x

Odd Function :

A Function is said to be an odd function if

f(-x)= -f(x)

Part a)

[tex]f(x)=x^3+x[/tex]

let us substitute x with -x

[tex]f(-x) = (-x)^3-x\\=-x \times -x \times -x\\=-x^3-x\\=-(x^3+x)\\=-f(x)[/tex]

Hence

f(-x)=-f(x)

There fore f(x) is an odd function

[tex]g(x)=x^2+1[/tex]

Substituting x with -x  we get

[tex]g(-x)=(-x)^2+1\\=-x \times -x+1\\=x^2+1\\=g(x)[/tex]

Hence g(-x)=g(x)

Therefore g(x) is an even Function.

Part b)

hf(x)=hx^3

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