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Let # f(x) # be the function # f(x) = 5^x - 5^{-x}. # Is # f(x) # even, odd, or neither ? Prove your result.

Answer 1

Theodore Amidon

The function is odd.

If a function is even, it satisfies the condition:

#f(-x)=f(x)#

If a function is odd, it satisfies the condition:

#f(-x)=-f(x)#

In our case, we see that

#f(-x)=5^-x-5^x=-(5^x-5^-x)=-f(x)#

Since #f(-x)=-f(x)#, the function is odd.

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Answer 2

Elijah Allen

The function ( f(x) = 5^x - 5^{-x} ) is an odd function.

Proof: To prove that ( f(x) ) is odd, we need to show that ( f(-x) = -f(x) ) for all ( x ) in the domain of ( f(x) ).

[ f(-x) = 5^{-x} - 5^{-(-x)} ] [ = 5^{-x} - 5^{x} ]

Now, compare ( f(-x) ) with ( -f(x) ): [ -f(x) = - (5^x - 5^{-x}) ] [ = -5^x + 5^{-x} ]

Since ( -f(x) = -5^x + 5^{-x} ) and ( f(-x) = 5^{-x} - 5^x ), we can see that ( f(-x) = -f(x) ).

Therefore, ( f(x) ) is an odd function.

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