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How do you find the compositions given #f(x)+3x-5# and #g(x)= sqrt(x-2)#?

Answer 1

Lily Anderson

#(f@g)(x)=3sqrt(x-2)-5#

#(g@f)(x)=sqrt(3x-7)#

By definition, #(f@g)(x)=f[g(x)]#

#=f(sqrt(x-2))#

#=3sqrt(x-2)-5#.

Similarly, #(g@f)(x)=g[f(x)]#

#=g(3x-5)#

#=sqrt(3x-5-2)#

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Answer 2

Liam Brennan

To find the composition ( (f \circ g)(x) ), substitute ( g(x) ) into ( f(x) ). So, ( (f \circ g)(x) = f(g(x)) ).

First, find ( g(x) ): ( g(x) = \sqrt{x - 2} ).

Now, substitute ( g(x) ) into ( f(x) ): ( f(g(x)) = f(\sqrt{x - 2}) ).

Finally, simplify ( f(\sqrt{x - 2}) ) by substituting ( \sqrt{x - 2} ) into ( f(x) ), which gives ( f(\sqrt{x - 2}) + 3\sqrt{x - 2} - 5 ).

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Answer from HIX Tutor

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.