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INSTRUCTORS Raffi Hovasapian John Zhu
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For more information, please see full course syllabus of Calculus AB
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Lecture Comments (6)

0 answers

Post by Osama Alajlan on December 8, 2012

your amazing thank you so much

0 answers

Post by Tanveer Sehgal on November 21, 2012

What is the use of an inverse function?

1 answer

Last reply by: Juan Melendez
Thu Mar 13, 2014 11:47 PM

Post by Hakyung Kim on July 19, 2012

In example 3, why does it become x times y^1/3 equals 2?

1 answer

Last reply by: Derrick Lu
Thu Jun 28, 2012 5:28 PM

Post by Callistus Elue on June 6, 2012

In example 2, Why does the 3 become - 3?

Related Articles:

Inverse Functions

  • Finding the inverse:
    • Switch x and y variables
    • Solve for y
    • Replace y with f—1(x)

Inverse Functions

Evaluate the following composite function h(x) = f(g(x)) f(x) where f(x) = cos(x + 3) and g(x) = x2 + 1
  • h(x) = cos(g(x) + 3) cos(x + 3)
  • = cos(x2 + 1 + 3) cos(x + 3)
= cos(x2 + 4) cos(x + 3)
Evaluate g(g(g(x))) where g(x) = 2x + 1
  • g(g(g(x))) = 2(g(g(x))) + 1
  • = 2(g(2x + 1)) + 1
  • = 2(2(2x + 1) + 1) + 1
  • = 2(4x + 2 + 1) + 1
  • = 8x + 6 + 1
= 8x + 7
Confirm that f(x) is the inverse of g(x) given f(x) = [(x + 3)/(x + 1)] and g(x) = [(3 − x)/(x − 1)]
f(x) is the inverse of g(x) when f(g(x)) = x
f(g(x)) = [([(3 − x)/(x − 1)] + 3)/([(3 − x)/(x − 1)] + 1)]
Multiplier the numerator and the denominator by x − 1
f(g(x)) = [(3 − x + 3x − 3)/(3 − x + x − 1)]
= [2x/2]
= x
Using the functions from the previous problem, find f(1) and g(2)
f(1) = [(1 + 3)/(1 + 1)]
= [4/2] = 2
g(2) = [(3 − 2)/(2 − 1)]
= [1/1] = 1
g(x) is the inverse of h(x). If g(y) = z, what is h(z)?
h(z) = y. This is a property of inverses. They ündo" changes made by their inverse function
Confirm that f(x) is the inverse of g(x) given f(x) = 10(x + 1) and g(x) = log10x − 1
g(f(x)) = log10(10(x + 1)) − 1
= (x + 1) − 1
= x
Find the inverse of the following function f(x) = [x/2] − 5
  • y = [x/2] − 5
  • swap x and y and solve for y
  • x = [y/2] − 5
  • x + 5 = [y/2]
  • y = 2(x + 5) = 2x + 10 = g(x)
  • Let's check our work and make sure it is an inverse
  • f(g(x)) = [(2x + 10)/2] − 5
  • = x + 5 − 5
  • = x
y = 2(x + 5) = 2x + 10 = g(x)
Find the inverse of the following function f(x) = lnx
  • x = lny
  • ex = elny
  • ex = y
The inverse is ex
Find the inverse of the following function f(x) = [((x + 1)[1/5])/2]
  • x = [((y + 1)[1/5])/2]
  • 2x = (y + 1)[1/5]
  • (2x)5 = y + 1
  • 32x5 − 1 = y
The inverse is 32x5 −1
Find the inverse of the following function f(x) = e2x
  • x = e2y
  • lnx = lne2y
  • lnx = 2y lne
  • lnx = 2y (1)
  • [lnx/2] = y
The inverse is [lnx/2]

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Inverse Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Inverse 0:08
    • Definition
    • Example: Finding the Inverse
  • Example 2 2:29
  • Example 3 3:12
  • Example 4 4:41