Find (fog)(2) and (gof)(-2)
f(x)=Square root of 4-x; g(x)=1-3x^2
Show work.
f(g) = sqrt(4-g) = sqrt(4-(1-3x^2)) = sqrt(3+3x^2)
f(g(2)) = sqrt(3+12) = sqrt(15)
or, g(2) = 1-12 = -11
f(-11) = sqrt(4+11) = sqrt(15)
do g(f) the same way.
To find (fog)(2), we need to first find g(2), and then substitute that value into f(x).
Step 1: Find g(2)
Substitute x = 2 into the equation g(x) = 1 - 3x^2:
g(2) = 1 - 3(2)^2
g(2) = 1 - 3(4)
g(2) = 1 - 12
g(2) = -11
Step 2: Substitute g(2) into f(x)
Substitute x = -11 into the equation f(x) = √(4 - x):
f(g(2)) = √(4 - (-11))
f(g(2)) = √(4 + 11)
f(g(2)) = √15
Therefore, (fog)(2) = √15.
To find (gof)(-2), we need to first find f(-2), and then substitute that value into g(x).
Step 1: Find f(-2)
Substitute x = -2 into the equation f(x) = √(4 - x):
f(-2) = √(4 - (-2))
f(-2) = √(4 + 2)
f(-2) = √6
Step 2: Substitute f(-2) into g(x)
Substitute x = √6 into the equation g(x) = 1 - 3x^2:
g(f(-2)) = 1 - 3(√6)^2
g(f(-2)) = 1 - 3(6)
g(f(-2)) = 1 - 18
g(f(-2)) = -17
Therefore, (gof)(-2) = -17.