Definitions: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows: Sum: (f + g)(x) = f (x) + g(x) Difference: (f – g)(x) = f (x) – g(x) Product: (f • g)(x) = f (x) • g(x) Quotient: (f / g)(x) = f (x)/g(x), provided g(x)  0 /

Example: Combinations of Functions Let f(x) = x2 – 3 and g(x)= 4x + 5. Find (f + g) (x) (f + g)(3) (f – g)(x) (f * g)(x) (f/g)(x) What is the domain of each combination?

Example: Combinations of Functions If f(x) = 2x – 1 and g(x) = x2 + x – 2, find: (f-g)(x) (fg)(x) (f/g)(x) What is the domain of each combination?

Review For Test 1 Equations of Lines Slope of a line, parallel & perpendicular Graph with intercepts, graph using slope-intercept method Formulae for distance and midpoint Graph circle with standard and general form (complete the square) Difference Quotient Piecewise-defined functions Domain & range of a function Intervals where the function increases, decreases, and/or is constant Be able to determine when a relation is a function – ordered pairs or graph Relative Max and Min Average Rate of change Even and odd functions, symmetry Transformations Algebra of functions, domain

The Composition of Functions The composition of the function f with g is denoted by f o g and is defined by the equation (f o g)(x) = f (g(x)). The domain of the composition function f o g is the set of all x such that x is in the domain of g and g(x) is in the domain of f.

Example: Forming Composite Functions Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f o g)(x) b. (g o f)(x) Solution We begin the composition of f with g. Since (f o g)(x) = f (g(x)), replace each occurrence of x in the equation for f by g(x). f (x) = 3x – 4 This is the given equation for f. (f o g)(x) = f (g(x)) = 3g(x) – 4 = 3(x2 + 6) – 4 = 3x2 + 18 – 4 = 3x2 + 14 Replace g(x) with x2 + 6. Use the distributive property. Simplify. Thus, (f o g)(x) = 3x2 + 14.

Example: Forming Composite Functions Given f (x) = 3x – 4 and g(x) = x2 + 6, find: a. (f o g)(x) b. (g o f)(x) Solution b. Next, (g o f )(x), the composition of g with f. Since (g o f )(x) = g(f (x)), replace each occurrence of x in the equation for g by f (x). g(x) = x2 + 6 This is the given equation for g. (g o f )(x) = g(f (x)) = (f (x))2 + 6 = (3x – 4)2 + 6 = 9x2 – 24x + 16 + 6 = 9x2 – 24x + 22 Replace f (x) with 3x – 4. Square the binomial, 3x – 4. Simplify. Thus, (g o f )(x) = 9x2 – 24x + 22. Notice that (f o g)(x) is not the same as (g o f )(x).