If the graph of functions 𝑓: 𝑅 → 𝑅 is symmetric to the graph of the function 𝑔: 𝑅 → 𝑅 with respect to the origin O of the Cartesian coordinate system Oxy , then for any real x

Question

If the graph of functions 𝑓: 𝑅 → 𝑅 is symmetric to the graph of the function 𝑔: 𝑅 → 𝑅 with respect to the origin O of the Cartesian coordinate system Oxy , then for any real x

a) g (x)= - f (x)

b) g (x )=f (-x )

c) g (x )=f( x)

d) g (x )= -f (-x)

Answer 1

Hi Irakli,

Symmetry with respect to the origin is a property of "odd" functions, which are functions defined as those satisfying the property f(-x) = - f(x). In other words, changing the sign of the input, changes the sign--and only the sign--of the output.

The wording of this problem makes it sound a little more complicated, in the sense that we are working with two functions, rather than a single odd function, but the idea is the same, if g(x) is symmetric to f(x), relative to the origin, then g(x) = -f(-x).

As an example, let f(x)=x, for x>=0, and g(x)=x for x<=0. Then g(-3)=-3=-f(3).

Regards,

John