Algebra/Chapter 11/Inverses of Functions

When functions were introduced in this chapter they were described as a special kind of relation. A relation is simply a connection between one number and some others. On the other hand a function has just one connection between a number and another number. For instance absolute value is a function because the absolute value of any number will map onto one, and only one number. On the other hand the square root operation is a relationship. ${\sqrt {x}}^{2}$ can be true for $x$ or $-x$.

We defined the identity value for addition as 0 and for multiplication as 1. This is because adding zero to a number does not change that number, or multiplying a number by 1 does not change that number. We defined the inverse of addition as subtraction. This is because a - a =0. The inverse of multiplication is division (except for 0) since a / a = 1. We define the inverse of a function as the function g such that g(f(x)) = x.

For example the inverse of the squared function is the square root function for whole numbers. That is for zero and non-negative rational numbers the ${\sqrt {x}}^{2}=x$. This is not true for the real numbers. For real numbers ${\sqrt {x}}^{2}=x\ or-x$.

When explaining the definition of domain and range of a relation we defined the vertical line test as stating that on a Cartesian plane if we let the x axis indicate the domain of a function and the y axis indicate the domain than the relationship is a function if the graph is not intersected more than once by a vertical line at every point on the graph. The vertical line test is an informal way to see if a relationship is a function. We can also use it to determine if the function has an inverse. For instance if we let $y=x^{2}$ we can see from its graph that it is a function. On the other hand if we graph $y={\sqrt {x}}$ we can see that a vertical line will cross the graph twice for every value except 0.