How do you find the compositions given #f(x)= sqrt(x2)# and #g(x)= x^21#?
The Function Composition tells us that:
where:
then we can think:
where:
then we can think:
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To find the composition of ( f(g(x)) ) and ( g(f(x)) ), first find ( f(g(x)) ) by substituting ( g(x) ) into ( f(x) ), and then find ( g(f(x)) ) by substituting ( f(x) ) into ( g(x) ).

Find ( f(g(x)) ): [ f(g(x)) = f(x^2  1) = \sqrt{x^2  1  2} = \sqrt{x^2  3} ]

Find ( g(f(x)) ): [ g(f(x)) = g(\sqrt{x  2}) = (\sqrt{x  2})^2  1 = x  2  1 = x  3 ]
So, the compositions are: [ f(g(x)) = \sqrt{x^2  3} ] [ g(f(x)) = x  3 ]
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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