Function Composition
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Transcript of Function Composition
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Function Composition
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Function Composition•Fancy way of denoting and performing SUBSTITUTION
•But first ….let’s review.
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•Function notation: f(x) This is read “f of x”.
•This DOES NOT MEAN MULITPLY f and x.
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Finding the Value of a Function
If f(x) = 3x - 1, find f(2).• Substitute 2 for x and simplify.• f(2) = 3(2) – 1 = 6 - 1
= 5
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Finding the Value of a Function
Given g(x) = x2 - x, find g(-3) g(-3) = (-3)2 - (-3) = 9 - -3 = 9 + 3 = 12 g(-3) = 12
Substitute -3 in for each x then simplify.
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Finding the Value of a Function
• Given g(x) = 3x - 4x2 + 2, find g(5)g(5) = 3(5) - 4(5)2 + 2 = 15 - 4(25) + 2 = 15 - 100 + 2 = -83
• g(5) = -83
Put 5 in place of every x.
Simplify.
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Find the Value of a Function
•Given f(x) = x - 5, find f(a+1) f(a + 1) = (a + 1) - 5 = a+1 - 5• f(a + 1) = a - 4
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Function Composition•Notation:
[fog](x) or f(g(x)) means “f of g of x”. [g0f](x) or g(f(x)) means “g of f of x”
The notation in orange is typically used.
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Find the Value of a Composition of Functions• Given f(x) = 2x + 5 and g(x) = 8 + x, find
f(g(-5)).• First find g(-5)…… g(-5) = 8 + -5 = 3.• Now put 3 in place of g(-5) (Plug 3 into the
f function) • f(g(-5)) = f(3) = 2(3) + 5
= 6 + 5 = 11
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Find the Value of a Composition of Functions
• Given f(x) = 2x + 5 and g(x) = 8 + x, find g(f(-5)). (Note how this problem is different from the problem on the previous slide.)
• First find f(-5) by plugging -5 in for x in the f function. f(-5) = 2(-5) + 5 = -10 + 5 = 5
• Now find g(f(-5)) by plugging 5 in for x in the g function.
g(5) = 8 + 5 = 13.• g(f(-5)) = 13
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.
a. f(g(3))
b. g(f(3))
c. f(g(0))
d. g(f(0))
e. g(g(3))
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions. Find f(g(3))
First find g(3). Remember this means find y when x = 3 for the g function.
When x = 3y = -1
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.
a. f(g(3))
First find g(3). Remember this means find y when x = 3.When x = 3
Now find f(-1).What is y when x = -1 for the f-function?
y = -1
When x = -1 y = -5So, f(g(3)) = -1
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.
Find: g(f(3))First find f(3).
What is y when x = 3 on the f function?
f(3) = 3
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.
Find: g(f(3))First find f(3).
What is y when x = 3 on the f function?
f(3) = 3Now find g(3).What is y when x = 3 on the g-function?g(3) = -1 So g(f(3)) = -1
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y = f(x)y = g(x)
Use the graphs of y = f(x) and y = g(x) to find each of the followingcompositions.
Find: f(g(0))
f(g(0)) = f(2) = 4
Find: g(f(0))
g(f(0)) = g(0) = 2
Find: g(g(3))
g(g(3)) = g(-1) = 3
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Composition of Functions• Function Composition is just fancy
substitution, very similar to what we have been doing with finding the value of a function.
• The difference is that instead of plugging in a number we will be plugging in another function.
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Composition of Functions• You still replace x with whatever is in
the parentheses.• f(g(x)) basically means plug the g
function into the f function and simplify.
• g(f(x)) basically means plug the f function into the g function and simplify.
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Composition of Functions• If f(x) = x2 + x and g(x) = x - 4, find f(g(x))• Plug the g function (x – 4) into the f function
in place of every x and simplify.• f(g(x)) = f(x - 4) = (x - 4)2 + x - 4
= x2 – 8x + 16 +x - 4 = x2 – 7x + 12
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Given f(x) = 2x + 2 and g(x) = x + 4, find f(g(x)).
f(g(x + 4)) = 2(x +4) +2 = 2x + 8 + 2
= 2x + 10 final answer
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If f(x) = x2 + x and g(x) = x – 4 find g(f(x)).
This time put the f-function into the g-function in place of x.
g(f(x)) =g(x2 + x ) = x2 + x – 4 Final answer.
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If f(x) = 2x + 2 and g(x) = x + 4, find g(f(x)).g(f(x)) = 2x +2 +4
= 2x + 6