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**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 /

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**Example: Combinations of Functions**

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

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**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?

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**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

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**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.

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**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 = 3x – 4 = 3x2 + 14 Replace g(x) with x2 + 6. Use the distributive property. Simplify. Thus, (f o g)(x) = 3x

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**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 = 9x2 – 24x + 22 Replace f (x) with 3x – 4. Square the binomial, 3x – 4. Simplify. Thus, (g o f )(x) = 9x2 – 24x Notice that (f o g)(x) is not the same as (g o f )(x).

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