1.12. Inverses of Functions http://www.ck12.org
What is the inverse off(x) =y=sinx?
Solution:The sine function does not pass the horizontal line test and so its true inverse is not a function.
However, if you restrict the domain to just the part of thex-axis between−π 2 andπ 2 then it will pass the horizontal
line test and the inverse will be a function.
The inverse of the sine function is called the arcsine function,f(x) =sin−^1 (x), and is shown in black. It is truncated
so that it only inverts a part of the whole sine wave. You will study periodic functions and their inverses in more
detail later.
Concept Problem Revisited
You can tell that two functions are inverses if each undoes the other, always leaving the originalx.
Vocabulary
A relationis a general term for functions and non-functions that relate two variables and may or may not pass the
vertical line test.
Two relations areinversesof each other if they are reflections across the liney=x.
Thehorizontal line testis a means of discovering whether the inverse of a function is also a function.
Guided Practice
- Determine the inverse off(x) = 5 +x 2. Verify that the inverse is actually the inverse.
- Determine iff(x) =^37 x−21 andg(x) =^73 x+21 are inverses of one another.