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Let $$h(x) = f(g(x)).$$ Note that $f(x)$ and $g(x)$ are not necessarily polynomials. State for each of the following cases whether $h(x)$ is even, odd, or neither.

 

a) $f(x)$ and $g(x)$ are both even.

b) $f(x)$ and $g(x)$ are both odd.

c) $f(x)$ is even and $g(x)$ is odd.

d) $f(x)$ is odd and $g(x)$ is even.

 Apr 3, 2018
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a) If both f and g are even, then \(h(x)=f(g(x))=f(g(-x))=h(-x)\)EVEN

b) If both and are odd, then \(h(-x)=f(g(-x))=f(-g(x))=-f(g(x))=-(h(x))\)ODD

c) If f is even and g is odd, then \(h(-x)=f(g(-x))=f(-g(x))=f(g(x))=h(x)\)EVEN

d) If f is odd and g is even, then \(h(x)=f(g(x))=f(g(-x))=h(-x)\)EVEN

 

smiley

 Apr 3, 2018

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