Operations on Functions 1.7 Operations on Functions Objectives: To find the domain of a function To find combinations of functions and their domains To find compositions of functions and their domains To determine original functions given their compositions

Finding Domains Algebraically Domain of a rational function: Find zeros of the denominator (factor if needed), Exclude zeroes from the domain – write intervals Practice (in your notes) pg. 219 your textbook – 

Finding Domains Algebraically Domain of a square root function: Set an expression under the square root greater or equal to zero (or just greater than), Solve an inequality and write intervals Practice (in your notes) pg. 219 your textbook –

Combinations of Functions

Combine like terms and write in standard form The sum f +g 1. To find the sum of two functions, substitute given functions, and simplify by combining like terms. Combine like terms and write in standard form

The difference f – g 2. To find the difference between two functions, subtract the first from the second. Make sure to distribute the “–” to each term of the second function. Simplify by combining like terms. Distribute negative

Write an answer in standard form The product f •g 3. To find the product of two functions, substitute the given functions, distribute to find the product, and then combine like terms. Write an answer in standard form

(Cannot be simplified here). The quotient f /g 4. To find the quotient of two functions, substitute the given functions, and then simplify/reduce if possible. (Cannot be simplified here).

The domain of combinations of functions: The domain of the sum, difference or product is the set of all numbers x in the domains of both f and g. For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0. Practice: In your notes – Pg. 220 # 39, 46

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

The Composition Function This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function.

This is read “g composition f ” and means to copy the g function down but where ever you see an x, substitute in the f function.

This is read “f composition f ” and means to copy the f function down but where ever you see an x, substitute in the f function. (Substitute the function into itself)

The DOMAIN of the Composition Function The domain of f composition g is the set of all numbers x in the domain of g such that g(x) (range of g) is in the domain of f. Range of g Domain of f Domain of g Range of f g f f (g(x)) = (f ₀ g)(x) X g(x)

The DOMAIN of the Composition Function The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. The domain of g is x  1 Denominator should not be = 0. This means that x  1, therefore, the domain of the composition would be combining the two restrictions.

In your notes – Pg. 220-221, # 54, 60, 62, 72-80 (even) Practice: In your notes – Pg. 220-221, # 54, 60, 62, 72-80 (even)