19 ## INVERSE FUNCTIONSThe graph of an inverse function THE INVERSE of a function reverses the action of that function. An elementary example are multiplication and division. Say that we begin with 5—and we multiply it times 3. 5 × 3 = 15. To undo that and return to 5, we must divide by 3. 15 ÷ 3 = 5. In the language of functions, let The function g( Actually, In general, if a function then if Here is the definition: Functions
for all values of
Problem 1. Let
what is the value of To see the answer, pass your mouse over the colored area.
For,
Example 1. Addition and subtraction are inverses. Subtracting a specific number reverses, or undoes, the result of adding it. In the language of functions, let
Upon applying the definition:
and
The definition is satisfied. The functions
Problem 2. Let To show that they are inverses, we must show that they satisfy the definition of inverses.
and
When we write ( then
Its inverse,
Example 2. Solve for (
To extract the argument of any function, simply take its inverse. In this example, we take the 4th root of both sides of the equation. We can immediately write
Therefore,
Problem 3. Solve for The inverse of taking the 5th root is taking the 5th power. Therefore, on taking the 5th power of both sides -- and thus freeing the argument:
The inverse of any function should be immediately clear. The inverse of In calculus, the student is expected to know those inverses. One sometimes sees that to "find" the inverse of a function, it is necessary to solve for Compare Problem 2. Say that we want to write the inverse of a this function:
Suppose that We would first multiply by 3 and then subtract 4. We would find 11. Therefore, to invert that and get back to 5, we would add 4 and then divide by 3.
That function is the inverse. The student should see and understand that.
Problem 4. a) Write the inverse of
Dividing by −5 is the inverse of multiplying by −5. b) Prove that they are inverses.
And
Problem 5. a) Let
For,
and then multiply by −2:
−2( b) Prove that
and
Problem 6. Let
The function As a notation for the inverse of a function
This is similar in
For the inverse trigonometric functions, see Topic 19 of Trigonometry. The graph of an inverse function The graph of the inverse of a function Reflect the graph about the (If we take the graph on the left to be the right-hand branch of To see that that is the graph of the inverse, let the graph of The figure on the left shows the reflection of We will see that the coördinates of
And
The definition of the inverse is satisfied. To see that the coördinates of But the angle at
Therefore the coördinates of So, when each point ( Each point ( Therefore we say that the graphs of a function and its inverse are symmetrical with respect to the straight line Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com |