Question

Let `f(x)= (x+3)/(x^2 -4x-12)` and `g(x)=sqrt(3-x)`, how do you find each of the compositions and domain and range?

Answer 1

As explained below.

Explanation:

`f@g(x) =` `(sqrt(3-x) +3)/((3-x)-4sqrt(3-x)-12)`

`g@f(x) =` `sqrt(3- (x+3)/(x^2-4x-12)`

f(x) can be written as `(x+3)/((x-6)(x+2))`,its domain would therefore be `(-oo,-2)U(-2,6)U(6,oo)`, or `{x in RR, x!= -2,6}`

For its range interchange x and y and solve the resultant quadratic equation for y. The result would be

y=`( (4x+1) +- sqrt((4x+1)(16x+1)))/(2x)`

This would mean `x<=-1/4` or `x>=-1/16`
Range would be`{y:RR, y<= -1/4 or y>= -1/16}`

For g(x), domain would be `{x:RR, x<=3}` so that all y values are real.

Range would be `{y: RR, y>=0}`