How do you determine if #f(x) = x^3 - 2# is an even or odd function?

Question

Sophie Adams · Accepted Answer

This function is neither even nor odd.

If a function is odd, #f(-x)=-f(x)# If a function is even, #f(-x) =f(x)# For #f(x)=x^3-2# #f(-x)=(-x)^3-2=-x^3-2# #f(-x) != -f(x)#, the function is not odd. #f(-x) !=f(x)#, the function is not even. The function is neither even nor odd. You could also examine the graph. Odd function are symmetric about the origin, i.e. they have the same shape in quadrants 1 and 3, or quadrants 2 and 4. Even function are symmetric about the y axis, i.e. the graph to the left of the y axis is the mirror image of the graph to the right of the y axis. If you examine the graph below, it matches neither of these descriptions.graph{x^3-2 [-10, 10, -5, 5]}

Lily Anderson · Answer

undefined To determine if $ f(x) = x^3 - 2 $ is an even or odd function, we evaluate $ f(-x) $ and compare it to $ f(x) $.

1. **Even Function**: If $ f(-x) = f(x) $ for all $ x $ in the domain of $ f $, then $ f(x) $ is an even function.
   
2. **Odd Function**: If $ f(-x) = -f(x) $ for all $ x $ in the domain of $ f $, then $ f(x) $ is an odd function.

Let's evaluate $ f(-x) $ for $ f(x) = x^3 - 2 $:

$$ f(-x) = (-x)^3 - 2 = -x^3 - 2 $$

Now, we compare $ f(-x) $ to $ f(x) $:

$$ f(-x) \neq f(x) $$
$$ f(-x) \neq -f(x) $$

Since neither condition for even nor odd functions is satisfied, $ f(x) = x^3 - 2 $ is neither an even nor an odd function.

Xavier Alonso · Answer

undefined To determine if a function is even or odd, we examine its symmetry properties.

1. Even Function:
A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in its domain.
For $ f(x) = x^3 - 2 $, if we substitute $ -x $ into the function, we get:
$ f(-x) = (-x)^3 - 2 = -x^3 - 2 $
If $ f(-x) $ equals $ f(x) $, then the function is even.
$ -x^3 - 2 $ is not equal to $ x^3 - 2 $ for all $ x $, so $ f(x) = x^3 - 2 $ is not an even function.

2. Odd Function:
A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $ in its domain.
For $ f(x) = x^3 - 2 $, if we substitute $ -x $ into the function, we get:
$ f(-x) = (-x)^3 - 2 = -x^3 - 2 $
If $ f(-x) $ equals $ -f(x) $, then the function is odd.
Since $ -x^3 - 2 = - (x^3 - 2) $, the function $ f(x) = x^3 - 2 $ is an odd function.