College Algebra : Symmetry

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #11 : Symmetry

Consider the function \displaystyle f(x)= x^{8}+ 2x^{4}+ 1.

Is \displaystyle f an even function, an odd function, or neither?

Possible Answers:

Even

Neither

Odd

Correct answer:

Even

Explanation:

A function \displaystyle f is even if, for each \displaystyle c in its domain,

\displaystyle f(-c)= f(c).

It is odd if, for each \displaystyle c in its domain,

\displaystyle f(-c)=- f(c).

Substitute \displaystyle -x for \displaystyle x in the definition:

\displaystyle f(-x)= (-x)^{8}+ 2(-x)^{4}+ 1

\displaystyle = (-1)^{8}x^{8}+ 2 (-1)^{4}x^{4}+ 1

\displaystyle = 1x^{8}+ 2 (1) x^{4}+ 1

\displaystyle = x^{8}+ 2x^{4}+ 1

\displaystyle = f(x)

Since \displaystyle f(x) = f(-x)\displaystyle f is an even function. 

Example Question #12 : Symmetry

\displaystyle f(x) is an even function. Let \displaystyle g(x) = \frac{f(x)}{x}

Is \displaystyle g an even function, an odd function, or neither?

Possible Answers:

Even

Odd

Neither

Correct answer:

Odd

Explanation:

A function \displaystyle f is even if, for each \displaystyle c in its domain,

\displaystyle f(-c)= f(c).

It is odd if, for each \displaystyle c in its domain,

\displaystyle f(-c)=- f(c).

Substitute \displaystyle -x for \displaystyle x in the definition of \displaystyle g:

\displaystyle g(-x) = \frac{f(-x)}{-x} = - \frac{f(-x)}{x}

Since \displaystyle f is even, \displaystyle f(-x)= f(x), so

\displaystyle g(-x) = - \frac{f(x)}{x} = -g(x )

This makes \displaystyle g an odd function. 

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