Chapter 3: Functions and Graphs 3.5: Operations on Functions

1. Chapter 3: Functions and Graphs3.5: Operations on Functions Essential Question: What is meant by the composition of functions?

2. 3.5: Operations on Functions • Just like integers or variables, functions can have operations performed on them. • Functions can be added or subtracted, multiplied or divided, or combined into a composite of functions. • Adding and subtracting functions works exactly as it sounds. • The sum function (f + g)(x) is simply f(x) + g(x) • Similarly, the difference function, (f – g)(x) is f(x) – g(x)

3. 3.5: Operations on Functions • Example 1: Sum and Difference Functions • For f(x) = and g(x) = • Write the rule for f + g and f – g • (f + g)(x) = • (f – g)(x) = • Remember, you can’t add/subtract terms underneath a square root (you can +/- the numbers in front if the roots are the same) • Find the domain of f + g and f – g • The domain of f is where 9 – x2> 0, that is • The domain of g is where x – 2 > 0, • The domain of the f + g and f – g consists of where the two domains meet, which would only be [-3, 3] [2, ∞) [2, 3]

4. 3.5: Operations on Functions • The product and quotient functions are similar. • Product function: • Quotient function:

5. 3.5: Operations on Functions • Example 2: Product and Quotient Functions • For f(x) = and g(x) = • Write the rule for • Find the domain of • The domain of f is where 3x > 0, or [0, ∞) • The domain of g is where x2 – 1 > 0, or (-∞, -1] and [1, ∞) • The domain of fg consists where the two domains meet, [1, ∞) • The domain of f/g is the same, with one additional rule: g ≠ 0(g is the denominator), so the domain is (1, ∞)

6. 3.5: Operations on Functions • Composition of functions • Another way of combining functions is to use the output of one function as the input of another. This operation is called composition of functions, and it’s denoted using the symbol ⁰ • (g ⁰ f)(x) == g(f(x)), meaning plug x into f(x) first. Use that answer to plug into g. • This is read “g of f of x”, or “g circle f” or “f followed by g” • ORDER MATTERS!!! Application is right to left.

7. 3.5: Operations on Functions • Example 3: Composite Functions • For f(x) = and g(x) = • Find the following: • (g ⁰ f)(2) • (f ⁰ g)(-1)

8. 3.5: Operations on Functions • Example 3: Composite Functions • For f(x) = and g(x) = • Find the following: • (g ⁰ f)(x) • (f ⁰ g)(x)

9. 3.5: Operations on Functions • Domains of Composite Functions • The domain of g ⁰ f is the set of all real numbers x such that: • 1) x is in the domain of f, and then • 2) f(x) is in the domain of g • The domain is the intersection of the inside function and the combination function • Example 4: If • Find g ⁰ f and f ⁰ g

10. 3.5: Operations on Functions • Find the domain of g ⁰ f and f ⁰ g • The domain of f is [0, ∞) • The domain of the composition is all real numbers • The domain for g ⁰ f is [0, ∞) • The domain of g is all real numbers • The domain of f is where x2 – 5 > 0, or (-∞, ] and [ , ∞) • The domain for f ⁰ g is (-∞, ] and [ , ∞)

11. 3.5: Operations on Functions • Assignment • Page 196-197 • 1-31, odd problems • Show work