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1.6 Inverse Functions

1.6 Inverse Functions. We can “undo” a function by reversing its domain and range. The new function we get by doing this is called the inverse function. Let f and g be two functions such that f ( g ( x )) = x for every x in the domain of g and

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1.6 Inverse Functions

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  1. 1.6Inverse Functions

  2. We can “undo” a function by reversing its domain and range. The new function we get by doing this is called the inverse function.

  3. Let f and g be two functions such that f (g(x)) = x for every x in the domain of g and g(f (x)) = x for every x in the domain of f. The function g is the inverse of the functionf Definition of the Inverse Function The Domain & Range values are “swapped.” A notation for inverse of f is f -1(x)

  4. Let f(x) contain these points: f -1(x) will contain these points:

  5. Ex. 1 Show that each function is the inverse of the other: f (x) = 5x and g(x) = x/5. To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)). f (x) = 5x f (g(x)) = 5( ) = 5(x/5) = x. Next, we find g(f (x)). g(x) = x/5 g(f (x)) = ( )/5 = 5x/5 = x. Notice how g(x) “undoes” the change produced by f(x).

  6. Ex. 2 Show that the functions are inverses:

  7. One important property of inverses is that they reflect over the line y = x

  8. Ex. 3 Are these functions inverses?

  9. Finding the Inverse of a Function • Replace f (x) by y in the equation for f (x). 2. Interchange x and y. 3. Solve for y. 4. Replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.

  10. Step 1 Replace f (x) by y. y = 7x – 5 Step 2 Interchange x and y. x = 7y – 5 Step 3 Solve for y. x + 5 = 7y x + 5 = y 7 Step 4 Replace y by f -1(x). x + 5 f -1(x) = 7 Ex. 2 Find the inverse of f (x) = 7x – 5.

  11. Ex. 3 Find the inverse of 1) 2) 3) 4)

  12. The Horizontal Line Test For Inverse Functions Not all functions have inverses. In order to have an inverse, it must be one-to-one: Every output corresponds to ONLY ONE input!

  13. How can you check for one-to-one? The Horizontal Line Test: A function f has an inverse f–1, if there is no horizontal line that intersects the graph of the function f at more than one point.

  14. Does f(x) = x2+3x-1 have an inverse function?

  15. This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.

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