Given that f(x) = x^2-3 and g(x) = 2x+1, find each of the following, if it exists.
a. (fg)(-root3)
b. (gf)(-1/2)
a. Determine what g(-sqrt3) is and call it x. Then calculate f(x)
x = g(-sqrt3) = -2 sqrt3 +1
f(x) = x^2 -3 = 12 -4 sqrt3 +1 - 3
= 10 - 4 sqrt3
Do b the same way.
This is as far as I got. Is this correct? Thanks!
g(f(x))
= g(x^2-3)
= 2(x^2-3) + 1
What you wrote for g(f(x))is correct. As a final step, they want you to substitute 1/2 for x.
I skipped the step of writing down the
g(f) function. You can start inserting -1/2 for x directly
f(-1/2) = (-1/2)^2 -3 = -2 3/4
= -11/4
g(f(1/2)) = -22/4 +1 = -18/4 = -4 1/2
Thanks for your help! I really appreciate it!
To find the values of (fg)(-√3) and (gf)(-1/2), we need to perform function composition.
a. (fg)(-√3):
First, we need to find the value of f(g(-√3)). Let's start by finding g(-√3):
g(x) = 2x + 1
g(-√3) = 2(-√3) + 1
= -2√3 + 1
Now, we substitute the value of g(-√3) into f(x) and evaluate f(-2√3 + 1):
f(x) = x^2 - 3
f(-2√3 + 1) = (-2√3 + 1)^2 - 3
= (4*3 - 4√3 + 1) - 3
= 12 - 4√3 + 1 - 3
= 10 - 4√3 - 2
= 8 - 4√3
Therefore, (fg)(-√3) = 8 - 4√3.
b. (gf)(-1/2):
First, we need to find the value of g(f(-1/2)). Let's start by finding f(-1/2):
f(x) = x^2 - 3
f(-1/2) = (-1/2)^2 - 3
= 1/4 - 3
= -11/4
Now, we substitute the value of f(-1/2) into g(x) and evaluate g(-11/4):
g(x) = 2x + 1
g(-11/4) = 2(-11/4) + 1
= -11/2 + 1
= -11/2 + 2/2
= -9/2
Therefore, (gf)(-1/2) = -9/2.