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AP Calculus AB: 6.3.5 Finding the Inverse of a Function

Mathematics12 CardsCreated 8 months ago

This content explains how to find the inverse of a function both algebraically and graphically by switching the input and output variables. It also emphasizes verifying the inverse through composition, ensuring that both directions return the original input value.

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Finding the Inverse of a Function

  • To determine the inverse of a function algebraically, swap the independent variable (x) and the dependent variable (y) and then solve for y.

  • Verify the inverse by composing it with the original function as described in the definition of an inverse.

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Key Terms

Term
Definition

Finding the Inverse of a Function

  • To determine the inverse of a function algebraically, swap the independent variable (x) and the dependent variable (y) and then solve for y...

note

  • To find the inverse of a function graphically, you
    reflect the curve of the function across the line given
    by y = x.

  • T...

Which of the following correctly relates the process of algebraically finding the inverse of a function f (x)?

To find the inverse of a function, first rename the function y, then swap y and x, and solve for y to get an expression. If both f −1 ( f (x)) = x ...

Given f (x) = e^x + 2x, which of the following is

f −1 (1)?

0

Given f(x)=2x+3/x−1, find f−1(x).

f−1(x)=x+3/x−2

Given f(x)=1/x+2+5, where x≠−2,find f−1(x).

f−1(x)=1/x−5 −2

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AP Calculus AB: 6.3.5 Finding the Inverse of a Function
TermDefinition

Finding the Inverse of a Function

  • To determine the inverse of a function algebraically, swap the independent variable (x) and the dependent variable (y) and then solve for y.

  • Verify the inverse by composing it with the original function as described in the definition of an inverse.

note

  • To find the inverse of a function graphically, you
    reflect the curve of the function across the line given
    by y = x.

  • This reflection just swaps the roles of y and x.

  • To find the inverse of a function algebraically, first
    rename the function y. Then swap y and the
    independent variable, which is usually x.

  • Solve for y to get an expression for the inverse
    function.

  • Since the original function is f ( x ), the inverse is
    noted as f^-1( x ).

  • In order for two functions to be inverses of each
    other, all roads must lead to x.

  • Evaluate the compositions f ( f^-1( x ) ) and f^-1( f ( x ) )
    to make sure they both equal x. You must check
    both directions.

Which of the following correctly relates the process of algebraically finding the inverse of a function f (x)?

To find the inverse of a function, first rename the function y, then swap y and x, and solve for y to get an expression. If both f −1 ( f (x)) = x and f ( f −1 (x)) = x, this expression is f −1 (x).

Given f (x) = e^x + 2x, which of the following is

f −1 (1)?

0

Given f(x)=2x+3/x−1, find f−1(x).

f−1(x)=x+3/x−2

Given f(x)=1/x+2+5, where x≠−2,find f−1(x).

f−1(x)=1/x−5 −2

To find the inverse of a one-to-one function f (x) graphically, reflect the function’s graph over ____________________.

y=x

Given f (x) = 4x^ 5 + 1, find f −1 (x).

f−1(x)=5√x−1/4

Given f (x) = x ^2, for x > 0, find f −1 (x).

f−1(x)=√x

Given ln(x−1)/3, find f−1(x).

f −1 (x) = e ^3x + 1

Given f (x) = ln (x^ 3 ), find f −1 (x).

f −1 (x) = e^ x / 3

Given f (x) = 2e ^3x + 8, find f −1 (x).

f−1(x)=1/3ln(x−8/2)