Download 1.4 Inverse Functions: Inverse Trigonometric Functions One-to

1.4
Inverse Functions: Inverse Trigonometric Functions
One-to-One Functions
A function which not only passes a vertical line test, but also passes a
horizontal line test.
For every x there corresponds only one y, and for every y there
corresponds only one x.
A function that is increasing over its domain is a one-to-one function. A
function that is decreasing over its domain is a one-to-one function.
Inverse Function of y = f(x)
To find the inverse of a one-to-one function y = f(x), interchange x and y,
and solve for y. The inverse function of y = f(x) is denoted by the symbol f
-1 (x).
The Domain of f is equal to the Range of f -1, and the Range of f is equal to
the Domain of f -1.
Also, f -1(f(x)) = f(f -1(x)) = x.
Example: Verify that the functions f(x) = 3 – 2x and g(x) = -½(x – 3) are
inverses.
Geometric Interpretation
The graph of a function f and the graph of its inverse f -1 are symmetric
with respect to the line y = x.
Example: For the function f(x) = 1 – 3x,
a)
find the inverse function f -1(x),
b)
find the domain and range of f and f -1, and
c)
graph f and f -1 on the same coordinate axes.
Inverse Trigonometric Functions
If
y = sin x
then
x is the angle whose sine is y.
That is,
x = arcsin y
So,
arcsin
2
 3 9 11 7
=
,
,
,
,
2
4 4 4
4
4
As we can see arcsin is not a function, because for one value of y, we get
multiple values of x. We can make arcsin a function by restricting its
domain and setting Arcsin x = Sin-1 x as follows:
Domain:
Range:
-1 < x < 1
-


< Sin-1 x <
2
2
Look at the graphs of y = arcsin x, y = Sin-1 x, and y = sin x on the same
coordinate system.
Example: Find the exact value of
2
2
a)
Sin-1
b)

3 
Sin-1  

 2 
c)
Sin-1 1
For arccos x, we can make it a function,
Arccos x = Cos-1 x = y
by the restriction
Domain:
-1 < x < 1
Range:
0 < Cos-1 x < 
Example: Find the exact value of
2
2
a)
Cos-1
b)

3 
Cos-1  

 2 
c)
Cos-1 1
For arctan x, we can make it a function,
Arctan x = Tan-1 x = y
by the restriction
Domain:
-∞ < x < ∞
Range:
-


< Tan-1 x <
2
2
Example: Find the exact value of
3
3
a)
Tan-1
b)
Tan-1  3
c)
Tan-1 1


For arcot x, we can make it a function,
Arccot x = Cot-1 x = y
by the restriction
Domain:
Range:
-∞ < x < ∞
0 < Cot-1 x < 
For arsec x, we can make it a function,
Arcsec x = Sec-1 x = y
by the restriction
Domain:
Range:
x≥1
or
x ≤ -1
0 ≤ Sec-1 x ≤ , y ≠

2
For arccsc x, we can make it a function,
Arccsc x = Csc-1 x = y
by the restriction
Domain:
Range:
x≥1
or
-
x ≤ -1


≤ Csc-1 x ≤ , y ≠ 0
2
2
Example: Find the exact value of
7 

Cos-1  cos

6 

Example: Find the exact value of
Tan-1 1u

Example: Find the exact value of sin Tan1 
3
2
Example: Find the exact value of cos(Sin-1 u)
Example: Evaluate Sec-1 (-5)
