**Definition.** A function f with domain D and range R is a **one-to-one
function **if

whenever x_{1} ≠ x_{2} in D; then f(x_{1}) ≠ f(x_{2}):

**Remark:** An equivalent way of writing the condition for a one-to-one
function is

this:

if f(x_{1}) = f(x_{2}); then x_{1} = x_{2} in D:

The arrow diagrams ilustrate a function that is one-to-one and a function
that is not

one-to-one.

Note that f never takes on the same value twice (any two numbers in A have
different

images), whereas g does take the same value twice (both 2 and 3 have the same
image,

4). In symbols, g(2) = g(3) but f(x_{1}) ≠ f(x_{2}) whenever x_{1} ≠ x_{2}.

**Horizontal line test:** A function f is one-to-one if no horizontal line
intersects the

graph of f in more than one point.

If a horizontal line intersects the graph of f at more than one point (as
shown in the

figure above), then there are numbers x_{1} ≠ x_{2} such that f(x_{1})
≠ f(x_{2}). This
means

that f is not one-to-one.

**Remark:** The horizontal line test is a geometric method for determining
whether a

function is one-to-one.

**Exercises.** Determine whether the function is one-to-one:

**Inverse function.**

**Definition.** Let f be a one-to-one function with domain A and range B. A
function

g with domain B and range A is the inverse of f provided that for every y in B
the

following is true:

g(y) = x if and only if f(x) = y:

The inverse function g is denoted by the symbol f^{-1}. Thus we have

domain of f^{-1} = range of f

range of f^{-1} = domain of f

f^{-1}(y) = x if and only if f(x) = y

The definition says that if f takes x into y, then f^{-1} takes y back into x.

**Remark: **Don't mistake the -1 in f^{-1} for an exponent. f^{-1}
**does not
mean** 1/f(x) .

The reciprocal 1/f(x) is written as (f(x))^{-1}.

**Example: **If f(1) = 5, f(3) = 7, and f(8) = -10, find f^{-1}(5), f^{-1}(7),
and f^{-1}(10).

Solution From the definition of f^{-1}, we have: f^{-1}(5) = 1, f^{-1}(7) = 3 and
f^{-1}(10) = 8.

**Theorem.** Property of Inverse Functions.

Let f be a one-to-one function with domain A and range B. If g is the inverse

function of f then the following conditions are true:

g(f(x)) = x for every x in A

f(g(y)) = y for every y in B:

Conversly, a function g satisfying these equation is an inverse of f.

Using the f^{-1} notation we have

f^{-1}(f(x)) = x for every x in A

f(f^{-1}(y)) = y for every y in B:

**Exercises.** Use the Property of Inverse Functions to show that f and g
are inverses

of each other:

(1) f(x) = x + 3 and g(x) = x - 3.

(2) f(x) = x^2 - 4 where x≥0 and g(x) =
where x≥-4.

(3) f(x) =
where x ≠ 1 and
where x ≠ 0.

**Finding the inverse of a function.**

Solve the equation y = f(x) for x in terms of y, obtaining an equation x =
f^{-1}(y).

To write the function f^{-1} as a function with argument x interchange x and y.

**Exercises .** Find the inverse of the function f.

**The relationship between the graphs of f and f ^{-1}.**

First note that b = f(a) is equivalent to a = f^{-1}(b). This implies that 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}. The

two graphs are re ections of each other through the line y = x, or are symmetric
with

respect to this line.

**Example: **(a) Sketch the graph of
(b) Use the graph of f to sketch

the graph of f^{-1}. (c) Find an equation for f^{-1}.