# Is the function f(x) = 13x^4 – 2x^3 + 7x even, odd or neither?

May 6, 2018

Since it is equal to neither, this function is neither odd or even

#### Explanation:

Perform a test:

For even functions:
$f \left(x\right) = f \left(- x\right)$

For odd functions:
$f \left(x\right) = - f \left(x\right)$

Negate the $x$ s in the equation and let's see:
f(x) = 13(-x)^4 – 2(-x)^3 + 7(-x)

$f \left(x\right) = 13 {\left(x\right)}^{4} + 2 {\left(x\right)}^{3} - 7 \left(x\right)$

Is this equal to:
$f \left(x\right)$, if not this is not an even function

$- f \left(x\right)$, if not this is not an odd function
$- f \left(x\right) = - 13 {\left(x\right)}^{4} + 2 {\left(x\right)}^{3} - 7 \left(x\right)$

Since it is equal to neither, this function is neither odd or even

May 6, 2018

neither

#### Explanation:

Given: $f \left(x\right) = 14 {x}^{4} - 2 {x}^{3} + 7 x$

If a function is even, it is symmetric about the $y -$ axis: $f \left(- x\right) = f \left(x\right)$

If a function is odd, it is symmetric about origin $y -$ axis: $f \left(- x\right) = - f \left(x\right)$

Assuming we don't have access to a graphing calculator, let's test for even #f(-x) = f(x):

original function: $f \left(x\right) = 14 {x}^{4} - 2 {x}^{3} + 7 x$

Replace $x$ by $- x$:
$f \left(- x\right) = 14 {\left(- x\right)}^{4} - 2 {\left(- x\right)}^{3} + 7 \left(- x\right)$

$f \left(- x\right) = 14 {x}^{4} + 2 {x}^{3} - 7 x \ne f \left(x\right) \text{ }$ Not an even function

$- - - - - - - - - - - - - - - - - - - -$
Test for an odd function, $f \left(- x\right) = - f \left(x\right)$:

original function: $f \left(x\right) = 14 {x}^{4} - 2 {x}^{3} + 7 x$

$- f \left(x\right) = - \left(14 {x}^{4} - 2 {x}^{3} + 7 x\right) = - 14 {x}^{4} + 2 {x}^{3} - 7 x$

Replace $x$ by $- x$ and $f \left(x\right)$ by $- f \left(x\right)$:
$- f \left(- x\right) = 14 {\left(- x\right)}^{4} - 2 {\left(- x\right)}^{3} + 7 \left(- x\right)$

$f \left(- x\right) = - \left(14 {x}^{4} + 2 {x}^{3} - 7 x\right)$

$f \left(- x\right) = - 14 {x}^{4} - 2 {x}^{3} + 7 x \ne - f \left(x\right)$