Chapter 1 | Functions and Graphs
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a. ⎛
⎞ ⎠ (3) = g ⎛
⎞ ⎠ = g (−2) =0
⎝ f (3)
⎝ g ∘ f
( g ∘ f )(0) = g (4) =5 b. The domain of g ∘ f is the set {−3, −2, −1, 0, 1, 2, 3, 4}. Since the range of f is the set {−2, 0, 2, 4}, the range of g ∘ f is the set {0, 3, 5}. c. ⎛ ⎝ f ∘ f ⎞ ⎠ (3) = f ⎛ ⎝ f (3) ⎞ ⎠ = f (−2) =4 ( f ∘ f )(1) = f ( f (1)) = f (−2) =4 d. The domain of f ∘ f is the set {−3, −2, −1, 0, 1, 2, 3, 4}. Since the range of f is the set {−2, 0, 2, 4}, the range of f ∘ f is the set {0, 4}.
Example 1.9 Application Involving a Composite Function
A store is advertising a sale of 20 % off all merchandise. Caroline has a coupon that entitles her to an additional 15 % off any item, including sale merchandise. If Caroline decides to purchase an item with an original price of x dollars, how much will she end up paying if she applies her coupon to the sale price? Solve this problem by using a composite function. Solution Since the sale price is 20 % off the original price, if an item is x dollars, its sale price is given by f ( x ) =0.80 x . Since the coupon entitles an individual to 15 % off the price of any item, if an item is y dollars, the price, after applying the coupon, is given by g ( y ) =0.85 y . Therefore, if the price is originally x dollars, its sale price will be f ( x ) =0.80 x and then its final price after the coupon will be g ( f ( x )) = 0.85(0.80 x ) =0.68 x .
1.6 If items are on sale for 10 % off their original price, and a customer has a coupon for an additional 30 % off, what will be the final price for an item that is originally x dollars, after applying the coupon to the sale price?
Symmetry of Functions The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function f ( x ) = x 4 −2 x 2 −3 shown in Figure 1.13 (a). If we take the part of the curve that lies to the right of the y -axis and flip it over the y -axis, it lays exactly on top of the curve to the left of the y -axis. In this case, we say the function has symmetry about the y -axis . On the other hand, consider the function f ( x ) = x 3 −4 x shown in Figure 1.13 (b). If we take the graph and rotate it 180° about the origin, the new graph will look exactly the same. In this case, we say the function has symmetry about the origin .
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