Addition of ordinates/ Composite functions Qs

[panicatthelunchbar]

Hi guys,

Can someone please explain how you would solve this, I get really confused?!

1. Given f: {x: x less than or equal 3} ---> R, where f(x) = 3-x and g: R-->R where g(x) = x^2 -1

a) Define a restricted function g* such that f(g(x))* is defined. Find f(g(x))*.

2. Let 'a' be a positive number such that f: [3. infinity)--->R where f(x) = a-x and g: (- infinity, 2] ---> R where g(x)=x^2 + a

a) Find all the values of 'a' for which f(g(x)) and g(f(x)) both exist.

Also, what is the easiest way to work with addition of ordinates?

Thanks

[TrueTears]

,   

So 1. we need to find so that any value of will only be a subset of

The function is a straight line and always decreasing, so the maximum value will be at its left endpoint, at . Hence, we can say . For it to have the correct range, we require

and 2. we need to find so that any value of will only be a subset of

For , determines the vertical translation of the standard parabola with vertex at (0,0). We can say . So to have the right range, .

Hence the answer is .