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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,

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Presentation on theme: "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,"— Presentation transcript:

1 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 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 /

2 Example: Combinations of Functions Let f(x) = x 2 – 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?

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

4 Review For Test 1 1.Equations of Lines 2.Slope of a line, parallel & perpendicular 3.Graph with intercepts, graph using slope-intercept method 4.Formulae for distance and midpoint 5.Graph circle with standard and general form (complete the square) 6.Difference Quotient 7.Piecewise-defined functions 8.Domain & range of a function 9.Intervals where the function increases, decreases, and/or is constant 10.Be able to determine when a relation is a function – ordered pairs or graph 11.Relative Max and Min 12.Average Rate of change 13.Even and odd functions, symmetry 14.Transformations 15.Algebra of functions, domain

5 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. 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.

6 Example:Forming Composite Functions Given f (x) = 3x – 4 and g(x) = x 2 + 6, find: a. (f o g)(x) b. (g o f)(x) Solution a.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(x 2 + 6) – 4 = 3x 2 + 18 – 4 = 3x 2 + 14 Replace g(x) with x 2 + 6. Use the distributive property. Simplify. Thus, (f o g)(x) = 3x 2 + 14.

7 Example:Forming Composite Functions 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) = x 2 + 6 This is the given equation for g. (g o f )(x) = g(f (x)) = (f (x)) 2 + 6 = (3x – 4) 2 + 6 = 9x 2 – 24x + 16 + 6 = 9x 2 – 24x + 22 Replace f (x) with 3x – 4. Square the binomial, 3x – 4. Simplify. Thus, (g o f )(x) = 9x 2 – 24x + 22. Notice that (f o g)(x) is not the same as (g o f )(x). Given f (x) = 3x – 4 and g(x) = x 2 + 6, find: a. (f o g)(x) b. (g o f)(x)


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