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

In summary, the conversation discusses the concept of function composition and how f(x) and f(-x) are different functions but are equivalent when a negative number is inputted. The conversation also mentions the idea of mirroring a function at the vertical axis and how this applies to all functions.
  • #1
dyn
773
62
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
 
Mathematics news on Phys.org
  • #2
dyn said:
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)##
 
Last edited:
  • Like
Likes dyn
  • #3
dyn said:
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.
 
  • Like
Likes dyn

FAQ: Are f(x) and f(-x) Equivalent Functions?

1. What is the argument of a function?

The argument of a function is the input or independent variable of a mathematical expression or equation. It is the value that is substituted into the function to produce an output or dependent variable.

2. How is the argument of a function represented?

The argument of a function is typically represented by a variable, such as x or y, or a combination of variables. It can also be represented by a specific value or set of values, depending on the context of the function.

3. Can the argument of a function be a complex number?

Yes, the argument of a function can be a complex number. In fact, many mathematical functions, such as trigonometric functions, have complex numbers as arguments.

4. What is the difference between the argument of a function and the domain of a function?

The argument of a function refers to the input or independent variable of a function, while the domain of a function refers to the set of all possible values that the argument can take. In other words, the domain is the range of values that the argument can be, while the argument itself is a specific value or variable.

5. How does the argument of a function affect its output?

The argument of a function directly affects its output. Changing the argument will result in a different output, as the function is essentially a rule or relationship between the argument and the output. In some cases, the argument can also be used to restrict the output of a function, such as in the case of a piecewise function.

Similar threads

Back
Top