LHS CAT4e ISM-04-final

# LHS CAT4e ISM-04-final - Chapter 4 INVERSE EXPONENTIAL AND...

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390 Chapter 4 INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS Section 4.1: Inverse Functions 1. Yes, it is one-to-one, because every number in the list of registered passenger cars is used only once. 2. It is not one-to-one because both Illinois and Wisconsin are paired with the same range element, 40. 3. This is a one-to-one function since every horizontal line intersects the graph in no more than one point. 4. This function is not one-to-one because there are infinitely many horizontal lines that intersect the graph in two points. 5. This is a one-to-one function since every horizontal line intersects the graph in no more than one point. 6. This function is one-to-one because every horizontal line will intersect the graph in exactly one point. 7. This is not a one-to-one function since there is a horizontal line that intersects the graph in more than one point. (Here a horizontal line intersects the curve at an infinite number of points.) 8. This function is not one-to-one because one horizontal line (the same line as the given graph) intersects the graph in infinitely many points. 9. 2 8 y x = Using the definition of a one-to-one function, we have ( ) ( ) 2 8 2 8 f a f b a b = = 2 2 a b a b = = . So the function is one-to- one. 10. 4 20 y x = + Using the definition of a one-to-one function, we have ( ) ( ) 4 20 4 20 f a f b a b = + = + 4 4 a b a b = = . So the function is one-to- one. 11. 2 36 y x = If 2 6, 36 6 36 36 0 0. x y = = = = = If x = - 6, ( ) 2 36 6 36 36 0 0. y = − − = = = Since two different values of x lead to the same value of y , the function is not one-to- one. 12. 2 100 y x = − If x = 10, 2 100 10 100 100 y = − = − 0 0 0. = − = − = If x = -10, ( ) 2 100 10 100 100 y = − − − = − 0 0 0. = − = − = Since two different values of x lead to the same value of y , the function is not one-to-one. 13. 3 2 1 y x = Looking at this function graphed on a TI-83, we can see that it appears that any horizontal line passed through the function will intersect the graph in at most one place. Another way of showing that a function is one- to-one is to assume that you have two equal y -values ( ) ( ) ( ) f a f b = and show that they must have come from the same x -value ( ) . a b = ( ) ( ) 3 3 3 3 3 3 3 3 3 3 2 1 2 1 2 2 f a f b a b a b a b a b a b = = = = = = So, the function is one-to-one. 14. 3 3 6 y x = Looking at this function graphed on a TI-83, we can see that it appears that any horizontal line passed through the function will intersect the graph in at most one place.

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Section 4.1: Inverse Functions 391 Another way of showing that a function is one- to-one is to assume that you have two equal y -values ( ) ( ) ( ) f a f b = and show that they must have come from the same x -value ( ) . a b = ( ) ( ) 3 3 3 3 3 3 3 3 3 3 3 6 3 6 3 3 f a f b a b a b a b a b a b = = = = = = So, the function is one-to-one. 15. 1 2 x y + = − Looking at this function graphed on a TI-83, we can see that it appears that any horizontal line passed through the function will intersect the graph in at most one place.
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