Let f(x) and g(x) be two odd functions defined for all real values of x. Given that s(x)=f(x)+g(x), prove that s(x) is also an odd function.

We recall that a function f(x) is said to be an odd function when f(-x)=-f(x).

We are told that f(x) and g(x) are odd functions, so we know from the above definition that:

1. f(-x)=-f(x)

2. g(-x)=-g(x)

Solution

We want to show that s(x) is an odd function. In other words, we want to show that s(-x)=-s(x) (that it satisfies the above definition).

We are told that s(x)=f(x)+g(x), so substituting x for -x, we get that

s(-x)=f(-x)+g(-x)

=-f(x)-g(x) (using 1 and 2)

=-(f(x)+g(x))

=-s(x) as required!

We have now shown that s(-x)=-s(x) and thus we have proven that s(x) is indeed an odd function.

Answered by Keir H. Maths tutor

12715 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Differentiate y=ln(ln(x)) with respect to x.


Given that (2x-1) : (x-4) = (16x+1) : (2x-1), find the possible values of x


What is the derrivative (dy/dx) of the equation 2 = cos 4x - cos 2y in terms of x and y?


Show that (sec(x))^2 /(sec(x)+1)(sec(x)-1) can be written as (cosec(x))^2.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences