Question

g 4. Determine which of the following functions are even, which are odd, and which are neither. (a) f(x) = x 3 + 3x (b) f(x) = 4 sin 2x (c) f(x) = x 2 + |x| (d) f(x) = e x (e) f(x) = 1 x (f) f(x) = 1 2 (e x + e −x ) (g) f(x) = x cos x (h) f(x) = 1 2 (e x − e −x ).

Answer 1

Answer:Given below

Explanation:

A function is said to be odd if

`F\left ( x\right )=F\left ( -x\right )`

(a)
`F\left ( x\right )=x^3+3x`

`F\left ( -x\right )=-x^3-3x=-\left ( x^3+3x\right )`

odd function

(b)
`F\left ( x\right )=4sin2x`

`F\left ( -x\right )=-4sin2x`

odd function

(c)
`F\left ( x\right )=x^2+|x|`

`F\left ( -x\right )=\left ( -x^2\right )+|-x|=x^2+|x|`

even function

(d)
`F\left ( x\right )=e^x`

`F\left ( -x\right )=e^(-x)`

neither odd nor even

(e)
`F\left ( x\right )=(1)/(x)`

`F\left ( -x\right )=-(1)/(x)`

odd

(f)
`F\left ( x\right )=(1)/(2)\left ( e^x+e^(-x)\right )`

`F\left ( -x\right )=(1)/(2)\left ( e^(-x)+e^(x)\right )`

even function

(g)
`F\left ( x\right )=xcosx(h)`

`F\left ( -x\right )=-xcosx(h)`

odd function

(h)
`F\left ( x\right )=(1)/(2)\left ( e^x-e^(-x)\right )`

`F\left ( -x\right )=(1)/(2)\left ( e^(-x)+e^(x)\right )`

odd function