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AP Calculus AB: 6.3.4 The Basics of Inverse Functions

Mathematics12 CardsCreated 8 months ago

This content introduces inverse functions, explaining how they reverse the effect of the original function by switching the input and output. It emphasizes graphical reflections over the line y = x and discusses the requirement for functions to be one-to-one in order to have inverses.

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The Basics of Inverse Functions

  • Inverse functions undo each other.

  • In inverse functions, the dependent variable and independent variable switch roles. The graph of an inverse function looks like a mirror reflection of the original graph.

  • Functions that are not one-to-one do not have inverses. One-to-one functions pass both the vertical line test and the horizontal line test.

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Key Terms

Term
Definition

The Basics of Inverse Functions

  • Inverse functions undo each other.

  • In inverse functions, the dependent variable and independent variable switch roles. The g...

note

  • A function f is like a machine that takes a number x and cranks out another number, f(x).

  • It can be helpful to have a machin...

The graph of an invertible fucntion, f(x),intersects with y=x at 22 points. At how many points will f intersect with f−1?

22

Which of these functions does not have an inverse?

This function is not invertible because its graph fails the horizontal line test.

Let f be an invertible function. Which of the following could be the graph of f^−1?

This graph has no trouble with the vertical and horizontal line tests.

The other graphs fail either the vertical or ...

Which of these is an incorrect statement regarding the function f (x) = 2x + sin x?

It is not possible to determine if f (x) is invertible.

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AP Calculus AB: 6.3.4 The Basics of Inverse Functions
TermDefinition

The Basics of Inverse Functions

  • Inverse functions undo each other.

  • In inverse functions, the dependent variable and independent variable switch roles. The graph of an inverse function looks like a mirror reflection of the original graph.

  • Functions that are not one-to-one do not have inverses. One-to-one functions pass both the vertical line test and the horizontal line test.

note

  • A function f is like a machine that takes a number x and cranks out another number, f(x).

  • It can be helpful to have a machine that reverses the process of the first machine. That machine is called the
    inverse function of the original function.

  • The inverse of a function f is noted by a raised –1. Do not confuse this with an exponent of –1, which symbolizes the reciprocal.

  • If you have a function that relates two variables, x and y, then the inverse function will switch them.

  • You can make the switch graphically by reflecting the first graph across the line given by y = x.

  • You can verify algebraically that f and f –1 are inverses of each other by composing them. Both f –1 (f(x)) and f(f –1 (x)) should equal x.

  • If the reflected image of a function does not pass the
    vertical line test, then it is not a function. Therefore the
    inverse does not exist.

  • You can see that if the curve of the original function (on the left) does not pass the horizontal line test, then its reflection (on the right) will not pass the vertical line test.

  • If a function is strictly increasing or strictly decreasing, then it is one-to-one.

The graph of an invertible fucntion, f(x),intersects with y=x at 22 points. At how many points will f intersect with f−1?

22

Which of these functions does not have an inverse?

This function is not invertible because its graph fails the horizontal line test.

Let f be an invertible function. Which of the following could be the graph of f^−1?

This graph has no trouble with the vertical and horizontal line tests.

The other graphs fail either the vertical or horizontal line test. This graph does not pass the horizontal line test.

Which of these is an incorrect statement regarding the function f (x) = 2x + sin x?

It is not possible to determine if f (x) is invertible.

If f (x) and g (x) are inverse functions of each other, which of these equations does not always hold?

f(x)g(x)=1

Let f(x) be invertible. Given the graph of f(x), which of the following depicts the graph of f^−1(x)?

This is the reflection over y = x.

Let f(x)=x^n, where −∞

n = 3

Let f(x) be invertible. Given the graph of f(x), which of the following is NOT true?

f^−1 is decreasing for x<0 and increasing for x>0

Which of the following is true for the function f(x) = e^2−x?

The function f (x) is invertible.

Let f(x) be an invertible function. If the graph of f(x) is given as follows, then f^−1(4) is equal to which of the following?

5