LHS CAT4e ISM-04-final

LHS CAT4e ISM-04-final - Chapter 4 INVERSE, EXPONENTIAL,...

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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. 28 yx =− Using the definition of a one-to-one function, we have () () 2 8 2 8 fa fb a b =⇒ = 22 aba b =⇒= . So the function is one-to- one. 10. 42 0 =+ Using the definition of a one-to-one function, we have ( ) ( ) 4 20 4 20 a b = ⇒+=+⇒ 44 b . So the function is one-to- one. 11. 2 36 If 2 6, 36 6 36 36 0 0. xy = = −= = 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 If x = 10, 2 100 10 100 100 y 00 0 . =− = If x = ±10, 2 100 10 100 100 y − − 0 . Since two different values of x lead to the same value of y , the function is not one-to-one. 13. 3 21 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 ( ) ( ) ( ) = and show that they must have come from the same x -value ( ) . ab = ( ) ( ) 33 333 3 3 3 a b b a ba b = = = = So, the function is one-to-one. 14. 3 36 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 ( ) ( ) ( ) fa fb = and show that they must have come from the same x -value ( ) . ab = ( ) ( ) 33 333 3 3 3 36 a b aba b a ba 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. We could also show that ( ) ( ) = implies . = ( ) ( ) 11 22 b a ++ = +=+⇒= So, the function is one-to-one.
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This note was uploaded on 11/13/2010 for the course MATH MAT 123 taught by Professor Smith during the Spring '10 term at Thomas Edison State.

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LHS CAT4e ISM-04-final - Chapter 4 INVERSE, EXPONENTIAL,...

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