Is the function #f(x) = 13x^4 – 2x^3 + 7x# even, odd or neither?
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Since it is equal to neither, this function is neither odd or even
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
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)#
