Functions: Operations and Composition

1. Unit 2: 2.8 Function Operations and Composition

2. Operations on Functions and Domain: • Sum function: (f + g)(x) = f(x) + g(x) • Difference function: (f – g)(x) = f(x) – g(x) • Product function: (fg)(x) = f(x) · g(x) • Quotient function: (x) = ; g(x) ≠ 0 • The domain of the sum, difference and product functions are all real numbers in the intersection of the domains of “f” and “g.” The domain includes the same for the quotient function, but also must exclude any values for which g(x) = 0.

3. Example 1: • Let f(x) = and g(x) = 3x + 5. Find the following: • a) (f + g)(1) b) (f – g)(-3) c) (fg)(5) d) (0) • Let f(x) = 8x – 9 and g(x) = . Find the following functions and determine their domains. • a) (f + g)(x) b) (f – g)(x) c) (fg)(x) d) (x)

4. Example 2: • If possible, use the given representations of functions “f” and “g” to evaluate: (f + g)(4) (f – g)(-2) (fg)(1) (0) • a) b) c) f(x) = 2x + 1 • g(x) =

5. Difference quotient • The difference quotient has to do with determining values of slope for a secant line as it approaches a linear graph at a certain point. • This will be discussed further in calculus. • The equation:

6. Example 3: • Let f(x) = . Find and simplify the expression for the difference quotient. • Let f(x) = . Find and simplify the expression for the difference quotient.

7. Composition of Functions & Domain • Composing a function is like putting one function into another function, then simplifying it to a new function. • Composition is noted as: f(g(x)) = ((x) • The domain of the composed function includes all numbers in the domain of “g” that also satisfies the domain of “f”

8. Example 4: • Let f(x) = 2x – 1 and g(x) = . • a) find (f )(2) b) Find (g f)(-3) • Given that f(x) = and g(x) = 4x + 2, find each of the following: • a) (f )(x) and its domain b) (g )(x) and its domain • Given that f(x) = and g(x) = , find each of the following: • a) (f )(x) and its domain b) (g )(x) and its domain

9. Example 5: • Let f(x) = 4x + 1 and g(x) = . Show that (g )(x) ≠ (f )(x).

10. Example 6: • Find functions “f” and “g” such that (f )(x) = • Find functions “f” and “g” such that (f )(x)= 4