Find the domain of the composite function f ₒ g . Please show all of your work.

f(x) = √x; g(x) = 2x + 12

f(x) = sqrt(x)

g(x) = 2x + 12
thus,
(fog)(x) = sqrt(2x + 12)
note that to get the domain, you must get the intersection of the 3 function domains.
for f(x) = sqrt(x), x cannot be less than zero thus the domain is [0 , +infinity)
for g(x) = 2x + 12, x can be any real number thus domain is (-infinity, +infinity).
for (fog)(x) = sqrt(2x + 12), the lowest possible value of the radicand (term inside the radical sign) is zero. thus equating 2x + 12 to zero,
2x + 12 = 0
x = -6
thus domain is [-6 , +infinity)

the intersection of the three domains is therefore,
[0, +infinity)

hope this helps~ :)

To find the domain of the composite function f ₒ g, we first need to determine the domain of g(x) and then substitute the resulting values into f(x).

Let's start by finding the domain of g(x), which is the set of all possible input values for g(x). In general, polynomials have a domain that includes all real numbers. Therefore, the domain of g(x) is (-∞, ∞), which means that g(x) can take any real value.

Now, we substitute the values of g(x) into f(x) to find the domain of f ₒ g.

The composition f ₒ g is written as f(g(x)), so we substitute g(x) (which is 2x + 12) into f(x):

f(g(x)) = √(2x + 12)

To determine the domain of f ₒ g, we need to find the values of x that make the expression √(2x + 12) defined.

For a square root function like √(2x + 12), the expression inside the square root (√(2x + 12)) must be greater than or equal to zero, since the square root of a negative number is not defined.

So we set 2x + 12 ≥ 0 and solve for x:

2x + 12 ≥ 0
2x ≥ -12
x ≥ -6

Therefore, the domain of f ₒ g is x ≥ -6, meaning that all real numbers greater than or equal to -6 are valid inputs for the composite function f ₒ g.