High School Are f(x) and f(-x) Equivalent Functions?

[dyn]

Hi.
If I have a function for example f( x ) = x² + x then to obtain f( -x) I just put (-x) in place of the x in f(x)
so I get f( -x) = x² - x
Am I right so far ?
So f(x) and f(-x) look like different functions but if you put a negative number in f(-x) it flips the -x back to +x so are f(x) and f(-x) are equivalent ?
But if I put sin(x) as the argument I get a totally different function

Thanks

 

[jbriggs444]

Hi.
If I have a function for example f( x ) = x² + x then to obtain f( -x) I just put (-x) in place of the x in f(x)
so I get f( -x) = x² - x
Am I right so far ?

Yes

So f(x) and f(-x) look like different functions

Yes. In fact, they are different functions. Which suggests that it is not a good idea to call them both by the same name. So invent a different name. Let g(x) = f(-x).

Now f(x) is a different function from g(x). In particular, f(1) = 1² + 1 = 2 and g(1) = f(-1) = (-1)² + (-1) = 0.

but if you put a negative number in f(-x) it flips the -x back to +x so are f(x) and f(-x) are equivalent ?

Yes, if you put a negative number into g(x) and the corresponding positive number into f(x) you get the same result out. The graphs of the two functions are mirror images of each other, reflected across the y axis.

g(x) = f(-x) and f(-x) = g(x).

But if I put sin(x) as the argument I get a totally different function

You lost me. What function are you getting when you do what to what and how is it different from what?

You may be talking about something called function composition where you apply two functions in sequence to the input argument. So if "-" is the name of the additive inverse function "-"(x) = -x then your original example could be thought of as the composition of f with "-": $g = f \circ -$ or g(x) = f(-x).

And your sin example could be thought of as the composition of f with sin: $g = f \circ \sin$ or $g(x) = f(\sin x)$

 

[mfb]

but if you put a negative number in f(-x) it flips the -x back to +x so are f(x) and f(-x) are equivalent ?

They are mirror versions of each other: If you mirror f(x) at the vertical axis you get f(-x). This is a general property of all functions (from real numbers to real numbers).
If f(x)=sin(x), then f(-x)=sin(-x)=-sin(x) which is the sine mirrored at the vertical axis.