Function Composition

Function Composition•Fancy way of denoting and performing SUBSTITUTION

•But first ….let’s review.

•Function notation: f(x) This is read “f of x”.

•This DOES NOT MEAN MULITPLY f and x.

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

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.

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.

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

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.

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

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

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.

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.

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

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

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.

If f(x) = 2x + 2 and g(x) = x + 4, find g(f(x)).g(f(x)) = 2x +2 +4

= 2x + 6