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Find the common domain and then find a formula for each of the functions f + g, f − g, f · g, fg . Solution. The domain of f (x) consists of all real numbers whereas the domain of g(x) consists of all numbers x ≥ 3. Thus, the common domain is the interval [−3, ∞). For any x in this domain we have √ (f + g)(x) = x + 1 + √x + 3 (f − g)(x) = x+1− √ x+3 √ (f · g)(x) = x x+3+ x+3 f g (x) = √x+1 x+3 provided x > −3. 2 Let f (x) = x2 − 3x + 2 and g(x) = 2x − 4. Evaluate the indicated function. (a) (f + g) 1 2 (b) (f − g)(−1) (c) (f g) 2 5 (d) f g (11).

A) f (x) = 2x + 4. (b) g(x) = x2 − 6. 4 Find f ◦ g and g ◦ f. (a) f (x) = 3x + 5, g(x) = 2x − 7. (b) f (x) = x3 + 2x, g(x) = −5x. 2 , g(x) = 3x − 5. (c) f (x) = x+1 √ 1 (d) f (x) = x2 , g(x) = x − 1. 3 (e) f (x) = |5−x| , g(x) = − x2 . 5 Evaluate each composite function where f (x) = 2x + 3, g(x) = x2 − 5x, and h(x) = 4 − 3x2 . (a) (f ◦ g)(−3) (b) (h ◦ g) 2 5 √ (c) (g ◦ f )( 3) (d) (g ◦ f )(2c). 64 6 Inverse Functions An important feature of one-to-one functions is that they can be used to build new functions.

6 In general, if c > 0, the graph of f (x) + c is obtained by shifting the graph of f (x) upward a distance of c units. The graph of f (x) − c is obtained by shifting the graph of f (x) downward a distance of c units. Horizontal Shifts This discussion parallels the one earlier in this section. Follow the same general directions. 7 Let f (x) = x2 . (a) Use a calculator to graph the function g(x) = (x + 1)2 = f (x + 1). How does the graph of g(x) compare to the graph of f (x)? (b) Use a calculator to graph the function h(x) = (x − 1)2 = f (x − 1).

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### A semester course in trigonometry by Marcel Finan

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