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Activity 28:

Activity 28:. Combining Functions (Section 3.6, pp. 269-275). The Algebra of Functions:. Let f and g be functions with domains A and B. We define new functions f + g, f − g , fg , and f/g as follows:. Example 1:. Let us consider the functions f(x) = x 2 − 2x and g(x) = 3x − 1.

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Activity 28:

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  1. Activity 28: Combining Functions (Section 3.6, pp. 269-275)

  2. The Algebra of Functions: Let f and g be functions with domains A and B. We define new functions f + g, f − g, fg, and f/g as follows:

  3. Example 1: Let us consider the functions f(x) = x2 − 2x and g(x) = 3x − 1. Find f + g, f − g, fg, and f/g and their domains:

  4. By the above the domain for f/g is all real numbers except when g(x) = 0. Consequently, the domain for f/g is:

  5. Example 2: Let us consider the functions : Find f + g, f − g, fg, and f/g and their domains:

  6. Consequently, the domain for f/g is Notice that

  7. Example 3: Use graphical addition to sketch the graph of f + g.

  8. Composition of Functions: Given any two functions f and g, we start with a number x in the domain of g and find its image g(x). If this number g(x) is in the domain of f, we can then calculate the value of f(g(x)). The result is a new function h(x) = f(g(x)) obtained by substituting g into f. It is called the composition (or composite) of f and g and is denoted by f ◦ g (read: ‘f composed with g’ or ‘f circle g’) That is we define: (f ◦ g)(x) = f(g(x)).

  9. Example 4: Use f(x) = 3x − 5 and g(x) = 2 − x2 to evaluate:

  10. Example 5: Find the functions f ◦ g, g ◦ f, and f ◦ f and their domains.

  11. But we also can’t have -1 in the domain Can’t have -1/2 in the domain

  12. Example 6: Express the function in the form F(x) = f(g(x)).

  13. Example 7: Find functions f and g so that f ◦ g = H if

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