43 the graph of y x 2 on the window 10 x 10 10 y 10

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43. The graph of y = x 2 on the window - 10 x 10 , - 10 y 10 appears identical (except for labels) to the graph of y = 2( x - 1) 2 + 3 if the latter is drawn on a graphing window centered at the point (1 , 3) with 1 - 5 2 x 1+5 2, - 7 y 13. 44. The graph of y = x 4 is below the graph of y = x 2 when - 1 x 1, and above it when x > 1. Both graphs have roughly the same up- ward parabola shape, but y = x 4 is flatter at the bottom. 45. p y 2 is the distance from ( x, y ) to the x -axis q x 2 + ( y - 2) 2 is the distance from ( x, y ) to the point (0 , 2). If we require that these be the same, and we square both quantities, we have y 2 = x 2 + ( y - 2) 2 y 2 = x 2 + y 2 - 4 y + 4 4 y = x 2 + 4 y = 1 4 x 2 + 1 In this relation, we see that y is a quadratic function of x . The graph is commonly known as a parabola. 46. The distance between ( x, y ) and the x-axis is p y 2 . The distance between ( x, y )and (1 , 4) is q ( x - 1) 2 + ( y - 4) 2 . Setting these equal and squaring both sides yields y 2 = ( x - 1) 2 + ( y - 4) 2 which simplifies to y = 1 8 ( x - 1) 2 + 16 (a parabola). 0.3 Inverse Fuctions 1. f ( x ) = x 5 and g ( x ) = x 1 / 5 f ( g ( x )) = f x 1 / 5 = x 1 / 5 5 = x g ( f ( x )) = g ( x 5 ) = ( x 5 ) 1 / 5 = x (5 / 5 ) = x 2. f ( x ) = 4 x 3 and g ( x ) = 1 4 x 1 / 3 f ( g ( x )) = 4 1 4 x 1 / 3 ! 3 = 4 1 4 x = x g ( f ( x )) = 1 4 4 x 3 1 / 3 = x 3. f ( x ) = 2 x 3 + 1 and g ( x ) = 3 r x - 1 2 f ( g ( x )) = 2 3 r x - 1 2 ! 3 + 1 = 2 x - 1 2 + 1 = x g ( f ( x )) = 3 r f ( x ) - 1 2 = 3 r 2 x 3 + 1 - 1 2 = 3 x 3 = x 4. f ( x ) = 1 x + 2 and g ( x ) = 1 - 2 x x f ( g ( x )) = 1 1 - 2 x x + 2 = 1 1 - 2 x x + 2 x x = x g ( f ( x )) = 1 - 2 1 x +2 1 x +2 = 1 - 2 1 x + 2 ( x + 2) = ( x + 2) - 2 = x 5. The function is one-to-one since f ( x ) = x 3 is one-to-one. To find the inverse function, write y = x 3 - 2 y + 2 = x 3 3 p y + 2 = x So f - 1 ( x ) = 3 x + 2 -1 1 0 -5 -2 4 3 5 4 x 0 -1 2 -4 1 -3 y -3 2 -2 -4 3 -5 5 6. The function is one-to-one with inverse f - 1 ( x ) = 3 x - 4
0.3. INVERSE FUCTIONS 15 -4 8 -8 4 2 y 0 -6 4 -10 6 0 -6 10 -2 10 -4 8 -2 x -8 2 6 -10 7. The graph of y = x 5 is one-to-one and hence so is f ( x ) = x 5 - 1. To find a formula for the inverse, write y = x 5 - 1 y + 1 = x 5 5 p y + 1 = x So f - 1 ( x ) = 5 x + 1 y 0 4 -5 -3 -3 4 3 5 5 2 -1 -2 1 -4 -5 -1 2 -2 0 3 -4 1 x 8. The function is one-to-one with inverse f - 1 ( x ) = 5 x - 4. 5 0 -5 2 -3 -3 2 5 4 x -4 -1 0 -2 1 -5 -1 -2 3 y 1 -4 3 4 9. The function is not one-to-one since it is an even function ( f ( - x ) = f ( x )). In particular, f (2) = 18 = f ( - 2). 10. Not one-to-one. Fails horizontal line test. 11. Here, the natural domain requires that the radicand (the object inside the radical) be nonnegative. Hence x ≥ - 1 is required, while all function values are non negative. Therefore the inverse, if defined at all, will be defined only for nonnegative numbers. Sometimes one can determine the existence of an inverse in the process of trying to find its formula. This is an example: Write y = p x 3 + 1 y 2 = x 3 + 1 y 2 - 1 = x 3 3 p y 2 - 1 = x The left side is a formula for f - 1 ( y ), good for y 0. Therefore, f - 1 ( x ) = 3 p x 2 - 1 when- ever x 0. 0 2 1 -2 -2 3 3 2 -1 x -1 y 1 0 12. Not one-to-one. Fails horizontal line test. 13. (a) Since f (0) = - 1, we know f - 1 ( - 1) = 0 (b) Since f (1) = 4, we know f - 1 (4) = 1 14. (a) Since f (0) = 1, we know f - 1 (1) = 0.