Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Algebra of Functions


• Section 2.6 in the textbook:
– Operations on Functions
– Difference Quotient
– Composition of Functions

Operations on Functions

• Functions can be added, subtracted,
multiplied, and divided to yield new
– Given two functions f(x) and g(x)
  • Sum: (f + g)(x) = f(x) + g(x)
  • Difference: (f – g)(x) = f(x) – g(x)
  • Product: (f · g)(x) = f(x) · g(x)
  • Quotient: (f / g)(x) = f(x) / g(x), g(x) ≠ 0

• Given two functions f(x) and g(x):
– The domain of their sum, difference, or
is the intersection of the domains of
f(x) & g(x)
– The domain of their quotient is the
of the domains of f(x) & g(x)
EXCLUDING those values of x such that
g(x) = 0

Operations on Functions (Example)

Ex 1: Let f(x) = x + 1 and g(x) = x^2 + x – 2, find:

a) f(x) + g(x) and then (f + g)(3)
b) f(x) – g(x)
c) f(x) · g(x)
d) f(x) / g(x)
Also, find the domain for each

Operations on Functions (Example)

Ex 2: Let and g(x) = x – 7, find:
f(x) / g(x) and its domain

Difference Quotient

• Given a function f, is called the
difference quotient
– Allows us to study how f changes as we allow x to
– You will see this again if you take Calculus
– For this class, just be able to calculate the difference
quotient and leave it in simplest form

Difference Quotient (Example)

Ex 3: Find the difference quotient of each

a) f(x) = 2x^2 – 3x + 1
b) f(x) = 5x – 2

Composition of Functions

• Given two functions f and g, the
composition of f and g is when g is used
as the input to f
–Written as (f ◦ g)(x) = f(g(x))
• This is NOT the same as f · g which is the product
– The domain of f ◦ g is the intersection of the
domain of g(x) and the domain of f(g(x))

Composition of Functions

Ex 4: Let f(x) = 2x – 7 and , find:
a) (f ◦ g)(x)
b) (g ◦ f)(x)
c) (f ◦ f)(x)
Also, find the domain of each

Ex 5: Let and , find:
a) (f ◦ g)(x)
b) (g ◦ f)(x)
Also, find the domain of each

Evaluating a Composition of

• To evaluate (f ◦ g)(a):
– Method I:
  • Compute (f ◦ g)(x)
  • Substitute x = a into the composition and evaluate
– Method II:
  • Compute b = g(a)
  • Compute f(b)

Evaluating a Composition of
Functions (Example)

Ex 6: Let f(x) = 4x + 3 and
g(x) = 3x^2 + x – 6, find:
a) (f ◦ g)(2)
b) (g ◦ f)(-1)
c) (f ◦ f)(½)


• After studying these slides, you should know
how to do the following:
– Add, subtract, multiply, or divide functions and
determine the domain of the new function
– Compute the difference quotient for a function
– Compute the composition of functions and be able to
compute its domain
– Evaluate a composition
• Additional Practice
– See the list of suggested problems for 2.6
• Next lesson
– Remainder & Factor Theorems (Section 3.1)