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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Solution (a) 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. $$f( -x) =( -x) ^{2}-5=x^{2}-5=f( x)$$

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.