Question

find any domain restrictions on f(g(x))
f(x)= square root x-3 and g(x)=x+5

Answer 1

Answer:

$x \ge -2$

Step-by-step explanation:

The "argument" of a square root (the quantity you're finding the square root of) must be greater than or equal to 0.

$f(x)=\sqrt{x-3}$ requires $x-3 \ge 0 \Rightarrow x \ge 3$.

But in the function f(g(x)), the input to  f  is g(x) = x + 5, so x + 5 must be greater than or equal to 3.

$x+5 \ge 3 \Rightarrow x\ge -2$

Composite functions can be confusing; one reason is the use of the same symbol, x to represent numbers from the domain of EACH function.  See if it helps to call the domain of g by the name x, and the domain of f by some other name.

$f(z)=\sqrt{z-3}\\g(x)=x+5\\f(g(x))=f(x+5)=\sqrt{(x+5)-3}=\sqrt{x+2}$

The square root will require $x+2 \ge 0 \Rightarrow x \ge -2$