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

Step-by-step 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)$F\left ( x\right )=\frac{1}{x}$

$F\left ( -x\right )=-\frac{1}{x}$

odd

(f)$F\left ( x\right )=\frac{1}{2}\left ( e^x+e^{-x}\right )$

$F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )$

even function

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

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

odd function

(h)$F\left ( x\right )=\frac{1}{2}\left ( e^x-e^{-x}\right )$

$F\left ( -x\right )=\frac{1}{2}\left ( e^{-x}+e^{x}\right )$

odd function