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.

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.

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 + 1 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

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 + 1 Find g(f(h(x))): 3x2 2(3x2) + 1 3(2x + 1)2 3(2(-5x) + 1)2

1.8 Combinations of Functions: Composite Functions III. Domains of Composite Functions. A) Domain restrictions (values that x cannot be). 1) Bottom ≠ 0 2) Inside √ cannot be negative. 3) Use this math to find the possible domain values for x. a) set bottom ≠ 0 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 #.

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, 31-37 all, 43, 45