1. 1.8 Combinations of Functions: Composite Functions
I. Sum, Difference, Product & Quotient of functions.
A) Sum: (f + g)(x) = f(x) + g(x)
B) Difference: (f - g)(x) = f(x) - g(x)
1) Distribute the negative sign into the g(x) terms
C) Product: (fg)(x) = f(x) g(x)
1) Distribute (FOIL) as needed.
D) Quotient: g(x) ≠ 0
1) Factor, cancel down & simplify as needed.

2. 1.8 Combinations of Functions: Composite Functions
II. Composition of Functions (f ◦ g)(x).
A) Composition: (f ◦ g)(x) = f(g(x))
B) Evaluating using a table [or ordered (x,y) pairs].
1) Create a table with x & g(x) [instead of y].
a) Plug in x values into g(x) and find the y.
2) Create a table with the g(x) values you found
[these are the new x values] & f(x) instead of y
a) Plug in the g(x) values you found into f(x) &
find the y.
b) This gives f(g(x)) for the given x value.

3. 1.8 Combinations of Functions: Composite Functions
II. Composition of Functions (f ◦ g)(x).
C) Simplifying using Substitution (book way).
1) Write as f(the g(x) function).
2) Plug the g(x) function into the x’s of f(x).
Examples:
Given: f(x) = 2x g(x) = 3x2
Find f(g(x)): f(3x2) = 2x + 1  2(3x2) + 1
Find (g ◦ f)(x): g(2x + 1) = 3x2  3(2x +1)2

4. 1.8 Combinations of Functions: Composite Functions
II. Composition of Functions (f ◦ g)(x).
D) Simplifying using Substitution (my way).
1) Write the function for the 1st function in (f ◦ g).
2) Replace all the x’s in that function with the
next function.
3) Repeat as needed.
Examples: Given f(x) = 2x + 1 , g(x) = 3x2 , h(x) = -5x
Find (f ◦ g)(x): 2x Find g(f(h(x))): 3x2
2(3x2) (2x + 1)2
3(2(-5x) + 1)2

5. 1.8 Combinations of Functions: Composite Functions
III. Domains of Composite Functions.
A) Domain restrictions (values that x cannot be).
1) Bottom ≠ ) Inside √ cannot be negative.
3) Use this math to find the possible domain values for x.
a) set bottom ≠ b) set inside √ > 0
B) Determine if any of these restrictions apply before
you simplify the composite function. They will still
apply after the math is done.
*C) If you have a factor on the bottom that cancels out
with a factor on the top, you have a special case.
1) There will be a “hole” in the graph at that #.

6. 1.8 Combinations of Functions: Composite Functions
IV. Finding Composite Function Values from a Graph.
A) Find the y coordinate of each function for the
given value of x.
1) Just look at the graph for the “y” value.
2) You can also do this from a table.
B) Do the math to the two “y” values as indicated.
1) Add them, Subtract them, Multiply them, etc.
HW page 89 # 1, 2, 5, 7, 9, 13 – 21 odd, all, 43, 45