Question

test whether the function is even or odd if f(x)=x³+4x+sin x​

Answer 1

Basic Concept :-

↝ Odd function

• A function f(x) is said to be odd if f(- x) = f(x)

↝ Even Function :-

• A function f(x) is said to be even if f(- x) = f(x)

Working Rule :-

↝ To check whether given function f(x) is even or odd,

• Step 1. Replace 'x' by '- x'

• Step 2. If f(- x) = f(x) then f(x) is even function or if f(- x) = - f(x), then f(x) is odd.

Let's solve the problem now!!

↝ Given function is

$\rm :\longmapsto\:f(x) = {x}^{3} + 4x + sinx$

So,

$\rm :\longmapsto\:f( - x) = {( - x)}^{3} + 4( - x) + sin( - x)$

$\rm :\longmapsto\:f( - x) = - {x}^{3} - 4x - sinx$

$\rm :\longmapsto\:f( - x) = - \: ({x}^{3} + 4x + sinx)$

$\bf\implies \:f( - x) = \: - \: f(x)$

$\bf\implies \:f(x) \: is \: odd \: function$

Answer 2

If a function f(x) is odd-function, f(-x) = - f(x)

If a function f(x) is even-function, f(-x) = f(x)

  In the question,

f(x) =x³ + 4x + sinx​

∴  f(-x) = (-x)³ + 4(-x) + sin(-x)

           = -x³ - 4x - sinx

           = - (x³ + 4x + sinx)

           = - f(x)

As f(-x) = - f(x),  this is an odd function.

Note that: sin(-x) = - sinx