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Math Help - Inverse Functions (recognizing)

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    Inverse Functions (recognizing)

    WE know how to graph four types of unctions including y=x, y=root of x, y=x^2, and y=1/x . However, we only had one lesson on INVERSE functions and then this homework which, in several questions, asks us to dtermine whether the inverse of a function is a function (poor wording on the book's behalf). The way we learned in determining this stuff is that, doing a vertical line test on a function proves it is a function, and doing a horizontal line test on an inverse function proves it is an inverse function. At the same time, we know the definition of a function to be "a rule that provides a single input for every valid input" and an inverse function... But what we can't quite do yet is recognize (simply based on the equation) is the inverse of a function is "a function" ... Now, like I said, the book gives poor wording so I don't quite understand what it means when it says to "find the inverse of a function and determine whether it is a function" , so perhaps who ever is reading htis can decide for themselves, but I am looking for an easier way to dterming whether an equation that the book gives IS an inverse function, or whether an equation the book gives IS a Function.... Simply by looking at the equation hopefully
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    Quote Originally Posted by mike_302 View Post
    WE know how to graph four types of unctions including y=x, y=root of x, y=x^2, and y=1/x . However, we only had one lesson on INVERSE functions and then this homework which, in several questions, asks us to dtermine whether the inverse of a function is a function (poor wording on the book's behalf). The way we learned in determining this stuff is that, doing a vertical line test on a function proves it is a function, and doing a horizontal line test on an inverse function proves it is an inverse function. At the same time, we know the definition of a function to be "a rule that provides a single input for every valid input" and an inverse function... But what we can't quite do yet is recognize (simply based on the equation) is the inverse of a function is "a function" ... Now, like I said, the book gives poor wording so I don't quite understand what it means when it says to "find the inverse of a function and determine whether it is a function" , so perhaps who ever is reading htis can decide for themselves, but I am looking for an easier way to dterming whether an equation that the book gives IS an inverse function, or whether an equation the book gives IS a Function.... Simply by looking at the equation hopefully
    I don't think there is a "simple" way to do it. You can either derive the form of the inverse function yourself or graph both functions and verify that they are reflections over the line y = x.

    To derive the form of the inverse function, take your given function in the form of y = f(x). The switch the x and y variables to get x = f(y) and solve for y. The result will only be a function if it passes the vertical line test.

    Another way to see if an inverse of a function is a function is to look at your given function and determine if it is 1 to 1. That is: for function y = f(x), given a value of y is there only one value of x? (We already know that given a value of x there is only one value of y, else f(x) is not a function.) If so then the inverse of f(x) will be a function.

    -Dan
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    so for any function were f(x)=x^2 or f(x)=sqrt (x) , the inverse of that function will NEVER be a function... IT will always be an inverse function... Is that correct?

    And, the horizontal line test only proves (assuming a "relation" passes it) that the "rule" is an inverse function.... IS that also correct? (Also understanding that a 1-1 function is a function that passes a horizontal AND vertical line test)

    (These types of statements help me understand inverse functions... Only reason I'm asking so, if I have part of the statement correct, but theres more to it such as exceptions, I would appreciate your input thanks)
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by mike_302 View Post
    so for any function were f(x)=x^2 or f(x)=sqrt (x) , the inverse of that function will NEVER be a function... IT will always be an inverse function... Is that correct?
    f(x) = x^2 has an inverse function if we restrict the domain to be  [0, \infty ), say. Then its inverse will be the function g(x) = \sqrt{x}. The inverse of f(x) = x^2 with a domain equal to the real line is not a function, thus does not exist. (Domains and ranges are critical to inverse functions. The domains, at least, should always be specified.)

    Quote Originally Posted by mike_302 View Post
    And, the horizontal line test only proves (assuming a "relation" passes it) that the "rule" is an inverse function.... IS that also correct? (Also understanding that a 1-1 function is a function that passes a horizontal AND vertical line test)
    Yes a 1 to 1 function passes both the vertical and horizontal line tests.

    -Dan
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