http://www.ck12.org Chapter 6. Transcendental Functions
6.1 Inverse Functions
Functions such as logarithms, exponential functions, and trigonometric functions are examples oftranscendental
functions.If a function is transcendental, it cannot be expressed as a polynomial or rational function. That is, it is
not analgebraic function.In this chapter, we will begin by developing the concept of an inverse of a function and
how it is linked to its original numerically, algebraically, and graphically. Later, we will take each type of elementary
transcendental function—logarithmic, exponential, and trigonometric—individually and see the connection between
them and their respective inverses, derivatives, and integrals.
Learning Objectives
A student will be able to:
- Understand the basic properties of the inverse of a function and how to find it.
- Understand how a function and its inverse are represented graphically.
- Know the conditions of invertibility of a function.
One-to-One Functions
A function, as you know from your previous mathematics background, is a rule that assigns a single value in its
range to each point in its domain. In other words, for each output number, there is one or more input numbers.
However, a function never produces more than a single output for one input. A function is said to be aone-to-one
function if each output is associated with only one single input. For example,f(x) =x^2 assigns the output 9 for both
3 and− 3 ,and thus it is not aone-to-onefunction.
One-to-One Function
The functionf(x)is one-to-one in a domainDiff(a) 6 =f(b),whenevera 6 =b.
There is an easy method to check if a function is one-to-one: draw a horizontal line across the graph. If the line
intersects at only one point on the graph, then the function is one-to-one; otherwise, it is not. Notice in the figure
below that the graph ofy=x^2 is not one-to-one since the horizontal line intersects the graph more than once. But
the functiony=x^3 is a one-to-one function because the graph meets the horizontal line only once.