Determine whether each of the following functions is even, odd, or neither. Then determine whether its graph is symmetric with respect to the \(y\)-axis, the origin, or neither.
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Since \(f( -x) =f( x) ,\) the function \(f\) is even. So the graph of \(f\) is symmetric with respect to the \(y\)-axis.
(b) The domain of \(g\) is \(\{ x|x\neq \pm \sqrt{5}\} \), so for every number \(x\) in its domain, \(-x\) is also in the domain. Replace \( x \) by \(-x\) and simplify. \[ g( -x) =\dfrac{4( -x) }{( -x) ^{2}-5}= \dfrac{-4x}{x^{2}-5}=-g( x) \]
Since \(g( -x) =-g( x) ,\) the function \(g\) is odd. So the graph of \(g\) is symmetric with respect to the origin.
(c) The domain of \(h\) is \(( -\infty ,\infty ) \), so for every number \(x\) in its domain, \(-x\) is also in the domain. Replace \(x\) by \(-x\) and simplify. \[ h( -x) = \sqrt[3]{5( -x) ^{3}-1}= \sqrt[3]{-5x^{3}-1}= \sqrt[3]{-( 5x^{3}+1) }=- \sqrt[3]{5x^{3}+1} \]
Since \(h( -x) \neq h( x) \) and \(h( -x) \neq -h( x) \), the function \(h\) is neither even nor odd. The graph of \( h\) is not symmetric with respect to the \(y\)-axis and not symmetric with respect to the origin.
(d) The domain of \(F\) is \(( -\infty ,\infty ) \), so for every number \(x\) in its domain, \(-x\) is also in the domain. Replace \(x\) by \(-x\) and simplify. \[ F( -x) =\vert {-}x\vert =\vert {-}1\vert \cdot \vert x\vert =\vert x\vert =F( x) \]
The function \(F\) is even. So the graph of \(F\) is symmetric with respect to the \(y\)-axis.
(e) The domain of \(H\) is \(\{ x|x\neq 5\}\). The number \(x=-5\) is in the domain of \(H\), but \(x=5\) is not in the domain. So the function \(H\) is neither even nor odd, and the graph of \(H\) is not symmetric with respect to the \(y\)-axis or the origin.