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Solutions and explanations for the concepts of inverse functions and logarithms in the context of calculus i. It covers the definition, properties, and graphs of inverse functions, as well as the relationship between inverse functions and logarithms. The document also includes examples and exercises.
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Inverse Functions
A function f is called a one-to-one function if it never takes on the same value twice; that is,
f (x 1 ) 6 = f (x 2 ) whenever x 1 6 = x 2.
Inverse Functions and Logarithms Inverse Functions
If a horizontal line intersects the graph of f in more than one point, then we see from Figure that there are numbers x 1 and x 2 such that f (x 1 ) = f (x 2 ). This means that f is not one-to-one.
Definition the same val
If a horizont from Figure 2 t that is not one mining whether
Horizontal Line intersects its
EXAMPLE 1 Is th
SOLUTION 1 If x 1
f
In the language of inputs and outputs, 1 this definition says that is one - to - one if each output corresponds to only one input.
f
0
y
⁄ ¤ x
fl ‡
y=ƒ
Is the function f (x) = x^3 one-to-one?
If x 1 6 = x 2 , then x^31 6 = x^32 (two diferent numbers can’t have the same cube). Therefore, by definition, f (x) = x^3 is one-to-one.
Inverse Functions and Logarithms Inverse Functions
From Figure we see that no horizontal line intersects the graph of f (x) = x^3 more than once. Therefore, by the Horizontal Line Test, f is one-to-one.
0 ⁄ ¤ x
This function is not one-to-one because f(⁄)=f(¤).
ƒ=˛ is one-to-one.
y
0 x
From Figure we see that there are horizontal lines that intersect the graph of g more than once. Therefore, by the Horizontal Line Test, g is not one-to-one.
and so
SOLUTION graph o one.
One possess
The
2
FIGURE 3 ƒ=˛ is one-to-one.
FIGURE 4
y=≈
Let f be a one-to-one function with domain A and range B. Then its inverse function f −^1 has domain B and range A and is defined by f −^1 (y) = x ⇔ f (x) = y
for any y in B.
For example, the inverse function of f (x) = x^3 is f −^1 (x) = x^1 /^3 because if y = x^3 , then f −^1 (y) = f −^1 (x^3 ) = (x^3 )^1 /^3 = x
Do not mistake the − 1 in f −^1 for an exponent. Thus f −^1 does not mean 1 /f.
The letter x is traditionally used as the independent variable, so when we concentrate on f −^1 rather than on f , we usually reverse the roles of x and y and write f −^1 (x) = y ⇔ f (y) = x. (1)
By substituting for y in Definition and substituting for x in (1), we get the following cancellation equations:
f −^1 (f (x)) = x x ∈ A f (f −^1 (x)) = x x ∈ B.
Find the inverse function of f (x) = x^3 + 2.
According to steps in previous slide, we first write
y = x^3 + 2
Then we solve this equation for x:
x^3 = y − 2 x = 3
y − 2
Finally, we interchange x and y:
y = 3
x − 2.
Therefore, the inverse function is f −^1 (x) = 3
x − 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 f. Since f (a) = b if and only if f −^1 (b) = a, the point (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 by reflecting about the line y = x.
the graph of. But we get the point . (See Figure 8.) Therefore, as illustrated by Figure 9:
FIGURE 8 0 y x (b, a) (a, b) y=x
f ^1 f ^1 b a a , b
8 CHAPTER 1 FUNCTIONS AND MODELS
The graph of f −^1 is obtained by reflecting the graph of f about the line y = x.
The graph of f −^1 is obtained by reflecting the graph of f about the line y = x.
0 x
y
As a check on our graph, notice that the expression for f −^1 is f −^1 (x) = −x^2 − 1 , x > 0. So the graph of f −^1 is the right half of the parabola y = −x^2 − 1 and this seems reasonable from Figure.
y
x
Trigonometry Angles
Angles
Angles can be measured in degrees or in radians (abbreviated as rad). The angle given by a complete revolution contains 360 ◦, which is the same as 2 π rad. Therefore π rad = 180◦^ (2) and 1 rad =
π
π 180 rad ≈ 0 .017 rad (3)
