One of the following functions is neither odd nor even. Which one?
x^5 + 3x
6x^2 – |x^2| + 3
x^9 + 3x^7 + 6x
(my answer)x^2 + 3x + 3 + |x –3|
This is really tricky pre-calc graphing problem.
Thanks for the help!
To determine whether a function is odd or even, we need to analyze its symmetry.
Let's start by explaining what it means for a function to be odd or even:
- An odd function is symmetric with respect to the origin. This means that if we reflect the graph of the function across the y-axis, it will look exactly the same as the original graph. Mathematically, an odd function follows the property f(-x) = -f(x).
- An even function is symmetric with respect to the y-axis. This means that if we reflect the graph of the function across the y-axis, it will still look exactly the same as the original graph. Mathematically, an even function follows the property f(-x) = f(x).
Let's analyze each of the given functions:
1. Function: x^5 + 3x
To determine if this function is odd or even, we need to check if the property f(-x) = -f(x) or f(-x) = f(x) holds true for this function.
Evaluate f(-x):
f(-x) = (-x)^5 + 3(-x) = -x^5 - 3x
Now let's compare f(-x) with -f(x):
-f(x) = -(x^5 + 3x) = -x^5 - 3x
Since f(-x) = -f(x), this function is odd.
2. Function: 6x^2 – |x^2| + 3
To determine if this function is odd or even, we need to check if the property f(-x) = -f(x) or f(-x) = f(x) holds true for this function.
Evaluate f(-x):
f(-x) = 6(-x)^2 – |-x^2| + 3 = 6x^2 – |x^2| + 3
Now let's compare f(-x) with -f(x):
-f(x) = -[6x^2 – |x^2| + 3] = -6x^2 + |x^2| - 3
Since f(-x) is not equal to -f(x), this function is neither odd nor even.
3. Function: x^9 + 3x^7 + 6x
To determine if this function is odd or even, we need to check if the property f(-x) = -f(x) or f(-x) = f(x) holds true for this function.
Evaluate f(-x):
f(-x) = (-x)^9 + 3(-x)^7 + 6(-x) = -x^9 - 3x^7 - 6x
Now let's compare f(-x) with -f(x):
-f(x) = -(x^9 + 3x^7 + 6x) = -x^9 - 3x^7 - 6x
Since f(-x) = -f(x), this function is odd.
4. Function: x^2 + 3x + 3 + |x – 3|
To determine if this function is odd or even, we need to check if the property f(-x) = -f(x) or f(-x) = f(x) holds true for this function.
Evaluate f(-x):
f(-x) = (-x)^2 + 3(-x) + 3 + |(-x) – 3| = x^2 - 3x + 3 + |x + 3|
Now let's compare f(-x) with -f(x):
-f(x) = -(x^2 + 3x + 3 + |x – 3|) = -x^2 - 3x - 3 - |x – 3|
Since f(-x) is not equal to -f(x), this function is neither odd nor even.
Therefore, the second function, 6x^2 – |x^2| + 3, is neither odd nor even.