Even and Odd Functions: why is (-x)^2 equal to (-1)^2 (x)^2 ?

I

Illvoices

Junior Member
Joined
Jan 13, 2017
Messages
116
If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x)=x2= is even because

Code: 
$$ f(-x)=(-x)[SUP]2[/SUP] = (-1)[SUP]2[/SUP]x[SUP]2[/SUP] = x[SUP]2[/SUP] = f(x) $$

can someone explain how'd the negative x squared went to be a negative one squared in this problem
 
Illvoices said:
If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x)=x2= is even because

Code: 
$$ f(-x)=(-x)[SUP]2[/SUP] = (-1)[SUP]2[/SUP]x[SUP]2[/SUP] = x[SUP]2[/SUP] = f(x) $$

can someone explain how'd the negative x squared went to be a negative one squared in this problem
Click to expand...

\(\displaystyle -x^2=(-1)(x)^2\) BUT \(\displaystyle (-x)^2=(-1)^2(x)^2=x^2\)
 
Even and Odd Functions

i wana know where did the -1 come from when the equation is x2​
 
Illvoices said:
i wana know where did the -1 come from when the equation is x2​
Click to expand...

With the given expression:

Can you calculate f(2)?

Can you calculate f(-2)?

Are those equal?

Try same with different numbers.
 
Illvoices said:
If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x)=x2= is even because

Code: 
$$ f(-x)=(-x)[SUP]2[/SUP] = (-1)[SUP]2[/SUP]x[SUP]2[/SUP] = x[SUP]2[/SUP] = f(x) $$

can someone explain how'd the negative x squared went to be a negative one squared in this problem
Click to expand...

You know that:

-x = (-1) * (x)

so

[-x]2 = [(-1) * (x)]2 →

[-x]2 = [(-1)]2 * [(x)]2 →

[-x]2 = [1] * [x2 ] →

[-x]2 = x2

Got it.....
 
Illvoices said:
can someone explain how'd the negative x squared went to be a negative one squared in this problem
Click to expand...

We have the following property of exponents.

(a*x)^n = a^n * x^n

In words, this says that, when a product is raised to an exponent, each factor gets raised to the exponent.

Example:

(2*x)^3 = 2^3 * x^3 = 8x^3


In your exercise, we have (-x)^2.

-x is a product: -1*x

Apply the property:

(-1*x)^2 = (-1)^2 * x^2 = x^2

:cool:
 
Illvoices said:
If a function f satisfies f(-x) = f(x) for every number x in its domain, then f is called an even function. For instance, the function f(x)=x2= is even because

Code: 
$$ f(-x)=(-x)[SUP]2[/SUP] = (-1)[SUP]2[/SUP]x[SUP]2[/SUP] = x[SUP]2[/SUP] = f(x) $$

can someone explain how'd the negative x squared went to be a negative one squared in this problem
Click to expand...
Just like -2 = (-1)*(2) we have -x = (-1)*x.

Now recall that (ab)2= a2b2 we have (-x)2 = [(-1)*x]2 = (-1)2*x2 = x2

Personally I would not have put the -1. I would simply say that (-x)2 = (-x)(-x) = x2 (using the fact that a neg times a neg is pos).
 
Last edited:
You must log in or register to reply here.
Top