Question

Consider the following functions:f(x) = cos(x3 - x)h(x) = |x-313g(x) = ln(x + 3)5(x) = sin(x)Which of the following is true?A. fand gare even, sis odd.B. hand s are odd, g is even.C. fis even, hand s are odd.D. sis odd, fand hare even.E. gand fare even, his odd.SUBMIT

Answer 1

The alternatives indicates that the question is aboud even and odd function.

An even function is one the gives:

`f(x)=f(-x)`

While an odd function is one tha gives:

`f(x)=-f(-x)`

The fisrt function is:

`\begin{gathered} f(x)=\cos ^3(x^3-x) \\ f(-x)=\cos ^3((-x)^3-(-x))=\cos ^3(-x^3+x)=\cos ^{}^(3)(-(x^3-x)) \end{gathered}`

Since

`\cos (x)=\cos (-x)`

Then

`\begin{gathered} \cos ^3(-(x^3-x))=\cos ^3(x^3-x) \\ f(x)=f(-x) \end{gathered}`

So, f is even.

We can do it similarly for g(x)

`\begin{gathered} g(x)=\ln (|x|+3) \\ g(-x)=\ln (|-x|+3)=\ln (|x|+3) \\ g(x)=g(-x) \end{gathered}`

For s(x), we have:

`\begin{gathered} s(x)=\sin ^(3)(x) \\ s(-x)=\sin ^(3)(-x) \end{gathered}`

Since:

`\sin (-x)=-\sin (x)`

Then:

`\begin{gathered} \sin ^3(-x)=(-\sin (x))^3=-\sin ^(3)(x) \\ s(x)=-s(-x) \end{gathered}`

So far we have f(x) even, g(x) even and s(x) odd. This is exactly what is said in alternative A:

A. f and g are even, s is odd.

so that is the right answer.