SOLUTION: Determine the symmetry (relative to the x-axis, y-axis, and/or origin) of the graph of the function. No need for graph. f(x)= x^3 + x

Algebra ->  Graphs -> SOLUTION: Determine the symmetry (relative to the x-axis, y-axis, and/or origin) of the graph of the function. No need for graph. f(x)= x^3 + x       Log On


   

Question 87302: Determine the symmetry (relative to the x-axis, y-axis, and/or origin) of the graph of the function. No need for graph. f(x)= x^3 + x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, a function is even when this equality is true:

f%28x%29=f%28-x%29

and a function is odd when this equality is true:

-f%28x%29=f%28-x%29

Lets see if the function is even:

y=x%5E3%2Bx Start with the given equation
Here's f%28x%29:
f%28x%29=x%5E3%2Bx This is the given equation
Here's f%28-x%29:
f%28-x%29=%28-x%29%5E3%2B%28-x%29 Replace x with -x

f%28-x%29=-x%5E3-x Simplify
Since x%5E3%2Bx does not equal -x%5E3-x, this means the equation f%28x%29=f%28-x%29 is not true.

Since f%28x%29 does not equal f%28-x%29, the equation y=x%5E3%2Bx is not an even function.



Now lets see if the function is odd:

y=x%5E3%2Bx Start with the given equation
Here's f%28-x%29:
f%28-x%29=-x%5E3-x Remember we solved for this previously
Here's -f%28x%29:
-f%28x%29=-%28x%5E3%2Bx%29 Negate the whole function

f%28-x%29=-x%5E3-x Distribute the negative and simplify
Since -x%5E3-x equals -x%5E3-x, this means the equation -f%28x%29=f%28-x%29 is true.

Since -f%28x%29 equals f%28-x%29, the equation y=x%5E3%2Bx is an odd function. This means the equation has symmetry with respect to the origin.

When we graph the equation y=x%5E3%2Bx, we can see if the equation has any symmetry:
+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E3%2Bx%29+
and we can clearly see that the equation is symmetrical with respect to the origin
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