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How do you write #y = 2 - sqrt(5 - (3x - 1)^2)# as a composition of two simpler functions?

Answer 1

To write ( y = 2 - \sqrt{5 - (3x - 1)^2} ) as a composition of two simpler functions, we can break it down into its constituent parts. The given function involves a square root operation nested within another operation. We can separate the inner function from the outer function.

Let's denote the inner function as ( u = 5 - (3x - 1)^2 ), and the outer function as ( y = 2 - \sqrt{u} ).

Now, we have two simpler functions:

Inner function: ( u = 5 - (3x - 1)^2 )
Outer function: ( y = 2 - \sqrt{u} )

By breaking down the original function into these two simpler functions, we can express it as the composition ( y = (2 - \sqrt{u}) ) where ( u = 5 - (3x - 1)^2 ).

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Answer 2

Liam Brennan

One way is to let #f(x)=2-sqrt(x)# and #g(x)=5-(3x-1)^2# so that #h(x)=2-sqrt(5-(3x-1)^2)=f(g(x))=(f circ g)(x)#.

There are many possible answers. Another answer is #f(x)=2-sqrt(5-x^2)# and #g(x)=3x-1# so that #h(x)=2-sqrt(5-(3x-1)^2)=f(g(x))=(f circ g)(x)#.

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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.