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INVERSE FUCTIONS in this lesson you will learn

INVERSE FUCTIONS in this lesson you will learn. What is an inverse function ? How do I find the inverse of a function ? Why do some functions have inverses and others do not ? How can I tell just from looking at a table or graph of a function whether that function will have an inverse?.

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INVERSE FUCTIONS in this lesson you will learn

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  1. INVERSE FUCTIONSin this lesson you will learn What is an inverse function? How do I find the inverse of a function? Why do some functions have inverses and others do not? How can I tell just from looking at a table or graph of a function whether that function will have an inverse?

  2. But first some important vocabulary! • FUNCTION: a relationship or expression involving one or more variables. "the function (bx + c)" • INVERSE OF A FUNCTION:  The relation formed when the independent variable is exchanged with the dependent variable in a given relation.  (This inverse may NOT be a function.) •  INVERSE FUNCTION:  If the above mentioned inverse of a function is itself a function, it is then called an inverse function. • One to One FUNCTION: A function for which every element of the range of the function corresponds to exactly one element of the domain.  Note: y = f(x) is a function if it passes the vertical line test. It is a 1-1 function if it passes both the vertical line test and the horizontal line test.

  3. What an inverse function is In mathematics, an inverse function is a function that "reverses" another function, (E.g) if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = y if and only if g(y) = x. Explanation: we all know that a function takes a starting value, performs some operation on this value, and creates an output answer right?So The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value. The inverse of function isrepresented by f-1(x). N.B The notation f -1(x) refers to "inverse function“ and does not algebraically mean 1/f (x).

  4. So how do we find the inverse of a function? There are 3 ways to find this inverse. • Swap ordered pairs • Solve algebraically • Graph

  5. Swap ordered pairs:  If your function is defined as a list of ordered pairs, all you have to do is swap the x and y values.  Remember, the inverse relation will be a function only if the original function is one-to-one. Example Given function f, find the inverse relation.  Is the inverse relation also a function? Answer: Function f  is a one-to-one function since the x and y values are used only once.  Since function f  is a one-to-one function, the inverse relation is also a function.Therefore, the inverse function is:

  6. Solvealgebraically:  Solving for an inverse relation algebraically is a three step process: • 1.  Set the function = y • 2.  Swap thex and yvariables • 3.  Solve for y Example Find the inverse of the function Remember:Set = y.Swap the variables.Solve for y. Use the inverse function notation since f (x) is a one-to-one function

  7. Graph:   The graph of an inverse relation is the reflection of the original graph over the identity line,y = x.  It may be necessary to restrict the domain on certain functions to guarantee that the inverse relation is also a function

  8. In some case you will need to adjust your answers • Square and Square Root • When we square a negative number, and then do the inverse, this happens: • But we didn't get the original value back! We got 2 instead of -2. • So the square function (as it stands) does not have an inverse • But we can fix that! • Restrict the Domain(the values that can go into a function). • All you have to do is make sure you don't use negative numbers. • In other words, restrict it to x ≥ 0so then we can have an inverse. • So imagine we have this situation: • x2 does not have an inverse • but {x2 | x ≥ 0 } • (which says "x squared such that x is greater than or equal to zero" usingset-builder notation)  • does in facthave an inverse. \

  9. Some functions don’t have inverses and here’s why To be able to have an inverse we need unique values. • Think about this… if there are two or more x-values for one y-value, how do we know which one to choose when goingback?.So we have this idea of "a unique y-value for every x-value", and it actually has a name. It is called "One-to-one"

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