# Combinations of Functions

## 1 Combinations of Functions, §1.5

**Sums, Differences, Products, and Quotients of Functions:** Let f and g
be two functions

with overlapping domains. Then, for all x common to both domains, the sum,
difference,

product, and quotient of f and g are defined as follows.

**Sum:** The domain is the intersection of the domains f and g.

(f + g) (x) = f (x) + g (x)

**Difference:** The domain is the intersection of the domains f and g.

(f − g) (x) = f (x) − g (x)

**Product:** The domain is the intersection of the domains f and g.

(f · g) (x) = f (x) · g (x)

**Quotient:** The domain is the intersection of the domains f and g, but
we must exclude the x’s

in g’s domain where g (x) = 0.

**1.1 Examples**

1. Given f (x) = 2x − 3 and g (x) = 3x^{2} + 2x − 5, find each of the following.

(a) The domain of f (x).

(b) The domain of g (x).

(c) f (x) + g (x) and its domain.

(d) f (x) − g (x) and its domain.

(e) f (x) · g (x) and its domain.

(f) f (x) /g (x) and its domain.

2. Given
find each of the following.

(a) The domain of f (x).

(b) The domain of g (x).

(c) f (x) + g (x) and its domain.

(d) f (x) − g (x) and its domain.

(e) f (x) · g (x) and its domain.

(f) f (x) /g (x) and its domain.

## 2 Composition of Functions, §1.5

The Composition: of the function f with the function g is

The domain of
is the set of all x in the domain of g such that g (x), i.e. the range of g, is

in the domain of f,

**2.1 Examples**

1. Given
find each of the following.

(a) The domain of f (x).

(b) The domain of g (x).

(c) The range of f (x).

(d) The range of g (x).

(e) and its domain.

(f) and its domain.

2. Given
and g (x) = x + 1, find each of the following. A graph of f is given

(following page) to help determine f’s range.

(a) The domain of f (x).

(b) The domain of g (x).

(c) The range of f (x).

(d) The range of g (x).

(e) and its domain.

(f) and its domain.

## 3 Inverse Functions, §1.6

**Definition:** A function f is called one-to-one if it never takes on the
same value twice; that is

whenever
.
We can often visually verified that a function is one-to-one by using the

horizontal line test, which states, “A function is one-to-one if an only if no
horizontal line

intersects its graph more than once. For example, the following function is not
one-to-one.

However, this function is one-to-one.

Some however may be mislead by the horizontal line test, because it is hard
to detect from

the visual alone that
is one-to-one. It should be clear though that f (x) =

is not one-to-one from the graph alone. To show that
is one-to-one

we proceed as follows. Since for all
which further

implies that
and finally that

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

function f^{−1} has a domain B and range A and is defined by

To find the inverse of a one-to-one function, follow these steps.

1. In the equation for f (x), replace f (x) by y.

2. Interchange the roles of x and y, and solve for y.

3. Replace y by f^{−1} (x) in the new equation.

4. Although not necessary, you may want to verify that

Let’s start with a simple example. Given a one-to-one
function, f (x) = 2x−1, find its domain,

range, and inverse.

**Work:**

First off we know f is one-to-one with domain x ∈R and range f (x) ∈R, so its
inverse, denoted

f^{-1} (x), has a domain x ∈R and range f^{-1} (x) ∈R. Using simple algebra (method
above) to

find f^{-1} (x).

You should observe the symmetry along the line y = x.

Here’s a more difficult example, however it is still
algebraically manageable. Using f (x) =

let’s find its inverse. First off we know f
is one-to-one with domain x≥−1 and range

f (x)≥0, so its inverse, denoted f^{-1} (x), has a domain x≥0 and range f^{-1} (x)≥
−1. Using

simple algebra to find f^{-1} (x).

You should observe the symmetry along the line y = x.

An even more difficult example, given,

find f^{-1} and its domain and range. A graph is provided as
a visual aid.

**Work:**

We finally have,

and its domain is (−1 1] and its range is