How do you determine whether the graph of #g(x)=(x^2-1)/x# is symmetric with respect to the origin?
The graph of g(x) is symmetric with respect to the origin
The graph of g(x) is symmetric with respect to the origin if
that's if g(x) is an odd function , then it is:
graph{(x^2-1)/x [-5, 5, -5, 5]}
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To determine whether the graph of ( g(x) = \frac{x^2 - 1}{x} ) is symmetric with respect to the origin, we can check if the function satisfies the condition of odd symmetry.
A function ( f(x) ) is said to be odd symmetric with respect to the origin if for every point ( (x, y) ) on the graph of ( f(x) ), the point ( (-x, -y) ) is also on the graph.
To check for odd symmetry, we evaluate ( g(-x) ) and compare it to ( -g(x) ):
[ g(-x) = \frac{(-x)^2 - 1}{-x} = \frac{x^2 - 1}{-x} = -\frac{x^2 - 1}{x} = -g(x) ]
Since ( g(-x) = -g(x) ), the function ( g(x) ) satisfies the condition for odd symmetry with respect to the origin.
Therefore, the graph of ( g(x) = \frac{x^2 - 1}{x} ) is symmetric with respect to the origin.
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To determine whether the graph of ( g(x) = \frac{x^2 - 1}{x} ) is symmetric with respect to the origin, you need to check if replacing ( x ) with ( -x ) in the equation results in the same function or its opposite.
[ g(-x) = \frac{(-x)^2 - 1}{-x} ]
If ( g(-x) = g(x) ), then the graph is symmetric with respect to the origin. If ( g(-x) = -g(x) ), then the graph is symmetric with respect to the origin as well.
Let's perform the substitution:
[ g(-x) = \frac{(-x)^2 - 1}{-x} ] [ = \frac{x^2 - 1}{-x} ] [ = -\frac{x^2 - 1}{x} ]
Since ( g(-x) = -g(x) ), the graph of ( g(x) = \frac{x^2 - 1}{x} ) is symmetric with respect to the origin.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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