Given f x x2 1 and gx 2x , find;
a) f2 b) g f2 c) g2 d) f g2 e) f gx
a) To find f(2), we substitute x = 2 into f(x) = x^2 + 1:
f(2) = (2)^2 + 1 = 4 + 1 = 5.
b) To find g∘f(2), we first find f(2) using the result from part a), which is f(2) = 5. Then we substitute this value into g(x):
g(f(2)) = g(5) = 2(5) = 10.
c) To find g(2), we substitute x = 2 into g(x) = 2x:
g(2) = 2(2) = 4.
d) To find f∘g(2), we first find g(2) using the result from part c), which is g(2) = 4. Then we substitute this value into f(x):
f(g(2)) = f(4) = (4)^2 + 1 = 16 + 1 = 17.
e) To find f∘g(x), we substitute g(x) = 2x into f(x):
f(g(x)) = f(2x) = (2x)^2 + 1 = 4x^2 + 1.
To find the value of each expression, we can substitute the given values into the functions.
a) To find f(2), substitute x = 2 into the function f(x):
f(2) = (2)^2 + 1
f(2) = 4 + 1
f(2) = 5
b) To find g∘f(2), we first find f(2), then substitute it into g(x):
f(2) = 5
g∘f(2) = g(f(2))
g∘f(2) = g(5)
g∘f(2) = 2 * 5
g∘f(2) = 10
c) To find g(2), substitute x = 2 into the function g(x):
g(2) = 2 * 2
g(2) = 4
d) To find f∘g(2), we first find g(2), then substitute it into f(x):
g(2) = 4
f∘g(2) = f(g(2))
f∘g(2) = f(4)
f∘g(2) = (4)^2 + 1
f∘g(2) = 16 + 1
f∘g(2) = 17
e) To find f∘g(x), substitute g(x) into f(x):
f∘g(x) = f(g(x))
f∘g(x) = f(2x)
f∘g(x) = (2x)^2 + 1
f∘g(x) = 4x^2 + 1
So, the values are:
a) f(2) = 5
b) g∘f(2) = 10
c) g(2) = 4
d) f∘g(2) = 17
e) f∘g(x) = 4x^2 + 1