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

2 Answers
May 6, 2018

Answer:

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

Explanation:

Perform a test:

For even functions:
#f(x)= f(-x)#

For odd functions:
#f(x)= -f(x)#

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

#f(x) = 13(x)^4 +2(x)^3 - 7(x)#

Is this equal to:
#f(x)#, if not this is not an even function

#-f(x)#, if not this is not an odd function
#-f(x)= -13(x)^4 + 2(x)^3 -7(x)#

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

May 6, 2018

Answer:

neither

Explanation:

Given: #f(x) = 14x^4 - 2x^3 +7x#

If a function is even, it is symmetric about the #y-# axis: #f(-x) = f(x)#

If a function is odd, it is symmetric about origin #y-# axis: #f(-x) = -f(x)#

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

original function: #f(x) = 14x^4 - 2x^3 +7x#

Replace #x# by #-x#:
#f(-x) = 14(-x)^4 - 2(-x)^3 +7(-x)#

#f(-x) = 14x^4 + 2x^3 -7x != f(x)" "# Not an even function

#--------------------#
Test for an odd function, #f(-x) = -f(x)#:

original function: #f(x) = 14x^4 - 2x^3 +7x#

#-f(x) = -(14x^4 - 2x^3 +7x) = -14x^4 + 2x^3 -7x#

Replace #x# by #-x# and #f(x)# by #-f(x)#:
#-f(-x) = 14(-x)^4 - 2(-x)^3 +7(-x)#

#f(-x) = -(14x^4 + 2x^3 -7x)#

#f(-x) = -14x^4 - 2x^3 +7x != - f(x)#