false

Class Notes
(835,798)

Canada
(509,410)

McGill University
(27,860)

Mathematics & Statistics (Sci)
(1,119)

MATH 140
(141)

Ewa Duma
(13)

Lecture 5

Unlock Document

Mathematics & Statistics (Sci)

MATH 140

Ewa Duma

Summer

Description

MATH140 - Lecture 5 Notes
Inverse Functions & Logs:
Deﬁnition: A function f(s) is a 1:1 function if it never takes on the same value force twice
Horizontal Line Test: A function is 1-to-1 if and only if no horizontal line intersects its
graph more than once
Example: f(x) = 5-4x^3
• Step 1: Replace f(x) for y
• Step 2: Solve for x
• Step 3: Interchange x for y
y = 5-4x^3
4x^3 = 5-y
x^3 = (5-y)/4
x = root3((5-y)/4) or x = ((5-y)/4)^(⅓)
Another way of understanding if a function f(x) is one to one is to see if we can ﬁnd its
inverse denoted by f^-1(x). To explain the inverse in simple terms, if f takes x to y then
f^-1 takes y to x.
f(x) = y 㲗 f^-1(y) = x
We note the following facts about inverse functions:
• F(f^-1(x)) = f^-1(f(x)) = x
• The graphs of f(x) and f^-1(x) are symmetric with respect to the line y = x
• Domain of f = Range of f^-1 and Range of f = Domain of f^-1
Deﬁnition: Let f be a 1-to-1 function with domain A and range B. It’s inverse function f^-1
is domain B and range A and is deﬁned by f^-1(y) = x 㲗 f(x) = y for any YEB
CAUTION: Do not mistake the -1 in f^-1 for an exponent. Thus, F^-1(x) does not mean
1/f(x)…
Example 2 (from textbook): Find the inverse function of f(x) = x^3 + 2 SECTION 1.6 INVERSE FUNCTIONS AND LOGARITHMS ||||
V EXAMPLE 4 Find the inverse function of f▯x▯ ▯ x ▯ 2 .
SOLUTION According to (5) we ﬁrst write
3
y ▯ x ▯ 2
Then we solve this equation for :x
3
x ▯ y ▯ 2
x ▯ s3y ▯ 2
Finally, we interchange x and :y
N In Example 4, notice hreverses the 3
effect of . The funcis the rule “Cube, y ▯ s x ▯ 2
then add 2”; is the rule “Subtract 2, then
take the cube root.” Therefore the inverse function is f▯1▯x▯ ▯ sx ▯ 2 .
The principle of interchanging x and y to ﬁnd the inverse function also gives us the
method for obtaining the graph of f▯1 from the graph of . Since f▯a▯ ▯ b if and only if
▯1
Logarithmic Functions:t ▯a, b▯is on the graph of f if and only if the point ▯b, a▯is on the
graph of f▯1 . But we get the point▯b, a▯from ▯a, b▯by reﬂecting about the line y ▯ x
64 |||| CHAPTER 1 FUNCTIONS AND MODELS Figure 8.)
If a > 0 and a ≠ 0, the exponential function f(x) = a^x is either increasing or decreasing
64 |||| CHAPTER 1 FUNCTIONS AND MODEyS (b,▯a) y
then we have and so is 1:1 by the Horizontal Line Test. Therefore, it has an inverse function which is
called the logthen we havection with base a shown by thf–!ollowing equation:
y
6 logax ▯ y &? a ▯ x,▯b)
0 6 logax ▯ y &? a ▯ x y
x x
Thus, if x ▯ 0 , thenlog x is the exponent to which the base a must be raised to givex. For
a y=x ▯3 y=x f
example, log 10001 ▯ ▯3 As an example,Thus,ifwx ▯ 0 , thenlog a01is the exponentto whichthe base muat be raisedto give . For
The cancellation equations (4), when applied to the functions f▯x▯ ▯ a x and
▯1 example, log 10001 ▯ ▯3 because 10 ▯ 0.001 .
f ▯x▯ ▯ log a , become We can see the two properties based on the following calculations:he functions
FIGURE 8 f▯1▯x▯ ▯ log x , become FIGURE 9

More
Less
Related notes for MATH 140

Join OneClass

Access over 10 million pages of study

documents for 1.3 million courses.

Sign up

Join to view

Continue

Continue
OR

By registering, I agree to the
Terms
and
Privacy Policies

Already have an account?
Log in

Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.