Given
f x 4 x
and
gx x x
2
find;
a)
f g1
b)
gf 1
c)
f f 5
d)
gg4
e)
f gx
f)
gf x
g)
f f x
h)
ggx
To find these compositions, we can substitute the given functions into each other and then evaluate at the specified values or expressions. Here are the steps for each one:
a) f ∘ g(-1):
First, substitute g(x) into f(x): f(g(x)) = f(x + x^2)
Then, substitute -1 into the resulting expression: f(g(-1)) = f((-1) + (-1)^2)
Simplify: f(g(-1)) = f(-1 + 1) = f(0) = 4 - 0 = 4
b) g(f(-1):
First, substitute f(x) into g(x): g(f(x)) = g(4 - x)
Then, substitute -1 into the resulting expression: g(f(-1)) = g(4 - (-1))
Simplify: g(f(-1)) = g(4 + 1) = g(5) = 5 + 5^2 = 5 + 25 = 30
c) f(f(-5)):
Substitute f(x) into f(x): f(f(x)) = f(4 - x)
Substitute -5 into the resulting expression: f(f(-5)) = f(4 - (-5))
Simplify: f(f(-5)) = f(4 + 5) = f(9) = 4 - 9 = -5
d) g(g(4)):
Substitute g(x) into g(x): g(g(x)) = g(x + x^2)
Substitute 4 into the resulting expression: g(g(4)) = g(4 + 4^2)
Simplify: g(g(4)) = g(4 + 16) = g(20) = 20 + 20^2 = 20 + 400 = 420
e) f(g(x)):
Substitute g(x) into f(x): f(g(x)) = f(x + x^2)
No further simplification can be done without knowing the specific value of x.
f) g(f(x)):
Substitute f(x) into g(x): g(f(x)) = g(4 - x)
No further simplification can be done without knowing the specific value of x.
g) f(f(x)):
Substitute f(x) into f(x): f(f(x)) = f(4 - x)
No further simplification can be done without knowing the specific value of x.
h) g(g(x)):
Substitute g(x) into g(x): g(g(x)) = g(x + x^2)
No further simplification can be done without knowing the specific value of x.
To find the given expressions, we will substitute the values into the respective functions and simplify the calculations. Let's solve each problem step-by-step:
a) f ∘ g(-1)
To find f ∘ g(-1), we first find g(-1) and then substitute the result into f(x).
Using the function g(x) = x + x^2, we can substitute x = -1:
g(-1) = (-1) + (-1)^2
= -1 + 1
= 0
Now, substituting the result into f(x), we have:
f(0) = 4 - 0
= 4
Therefore, f ∘ g(-1) = 4.
b) g(f(-1))
To find g(f(-1)), we first find f(-1) and then substitute the result into g(x).
Using the function f(x) = 4 - x, we can substitute x = -1:
f(-1) = 4 - (-1)
= 4 + 1
= 5
Now, substituting the result into g(x), we have:
g(5) = 5 + 5^2
= 5 + 25
= 30
Therefore, g(f(-1)) = 30.
c) f(f(-5))
To find f(f(-5)), we need to find f(-5) first and then substitute the result into f(x).
Using the function f(x) = 4 - x, we can substitute x = -5:
f(-5) = 4 - (-5)
= 4 + 5
= 9
Now, substituting the result into f(x), we have:
f(9) = 4 - 9
= -5
Therefore, f(f(-5)) = -5.
d) g(g(4))
To find g(g(4)), we need to find g(4) first and then substitute the result into g(x).
Using the function g(x) = x + x^2, we can substitute x = 4:
g(4) = 4 + 4^2
= 4 + 16
= 20
Now, substituting the result into g(x), we have:
g(20) = 20 + 20^2
= 20 + 400
= 420
Therefore, g(g(4)) = 420.
e) f(g(x))
To find f(g(x)), we substitute the expression g(x) into the function f(x):
f(g(x)) = 4 - (x + x^2)
f) g(f(x))
To find g(f(x)), we substitute the expression f(x) into the function g(x):
g(f(x)) = f(x) + (f(x))^2
g) f(f(x))
To find f(f(x)), we substitute the expression f(x) into the function f(x):
f(f(x)) = 4 - f(x)
h) g(g(x))
To find g(g(x)), we substitute the expression g(x) into the function g(x):
g(g(x)) = g(x) + (g(x))^2