Please Help! Confused! I got an answer but it is not correct. Can someone help? Thanks!

Given that f(x) = x^2 - 3 and g(x) = 2x + 1, find each of the following, if it exists.

(gf) (-1/2)

f(-1/2) = 1/4 - 12/4 = -11/4

g(-11/4) = 2(-11/4) + 1
= -11/2 + 2/2
= -9/2

How do we get this? and do they both exist?

Of course! I'd be happy to help you with that. To find the composite function (gf)(x), we need to substitute the function g(x) into the function f(x) and evaluate it at a specific value, -1/2 in this case.

Now let's proceed step by step. First, we substitute g(x) = 2x + 1 into f(x):

f(x) = x^2 - 3

Substituting g(x) = 2x + 1:

f(g(x)) = (2x + 1)^2 - 3

Now we can evaluate this composite function at x = -1/2. To do this, we substitute x with -1/2:

f(g(-1/2)) = (2(-1/2) + 1)^2 - 3

Simplifying the expression inside the parentheses:

f(g(-1/2)) = (-1 + 1)^2 - 3

Since -1 + 1 = 0, we can further simplify the expression:

f(g(-1/2)) = 0^2 - 3

Now we can evaluate the equation:

f(g(-1/2)) = 0 - 3

Finally, performing the subtraction:

f(g(-1/2)) = -3

Thus, the value of the composite function (gf)(-1/2) is -3.