Topic 4-1: Algebra of Functions Do mains of Created

Topic 4-1: Algebra of Functions New functions can be created by adding, subtracting, multiplying, or dividing existing functions.

Sum of Functions: y f x g x

Difference of Functions: y f x g x

Product of Functions: y f x g x

Quotient of Functions:

f xy

g x

Domains of Created Functions Let A be the domain of f and B be the domain of g. New functions created by addition, subtraction, or multiplication will have a domain of .A B

New functions created by the division will have a domain of

0 .A B x g x

If two functions with unlimited domains over all real numbers are added, subtracted, or multiplied, the newly created function will have an unlimited domain over all real numbers.

Determine the domain of the given function by finding the domains of the component functions and using the algebra of functions.

Ex. 1 2 2y x

x

Ex. 2 31y x x

Determine the domain of the given function by finding the domains of the component functions and using the algebra of functions.

Ex. 3 1 1y x x

Ex. 4

2

3

xy

x

Write the new function given functions f and g. Determine the domain of the new function.

Ex. 1 ( )f x x 2

( )g x x

a. f + g

b. f · g

c. f / g

Write the new function given functions f and g. Determine the domain of the new function.

Ex. 2 ( ) 1f x x

1( )

1g x

x

a. f – g

b. f · g

c. f / g

Topic 4-2a: Composition of Functions Unlike the algebra of functions, which creates new functions by adding or multiplying them together, compositions of functions creates new functions by putting one function inside another function. The composition of functions f and g is

denoted f g and defined as ( ) .f g x f g x

Domains of Compositions Let A be the domain of g and B be the domain of f. Then the domain of a composition of functions ( )f g is found by:

dom ain of ( )

is in but is not in

f g

A x x A g x B

As with the algebra of functions, if two functions with unlimited domains over all real numbers are composed, their composition will have an unlimited domain as well.

Ex. 1a Find ( ),f g and its domain, if

2

3 4 and 2 .f x x g x x x

Ex. 1b Find ( )g f if

2


Ex. 1c Find ( )f f

if 2


Ex. 2 Find ( ), ( ), and ( ),f g g f g g and each

function’s domain, if

2

2 1 and 4 1.f x x g x x

Ex. 3 Find ( ) and ( ),f g g f and each

function’s domain, if

2

1 and 1.f x x g x x

Topic 4-2b: Decomposing Functions

Find the functions f and g such that .y f g x

Ex. 1: 2

2

3xy

x

Ex. 2: 4

2 3y x

Find the functions f and g such that .y f g x

Ex. 3: 2 3y x

Ex. 4: 2

2 21 2 1y x x

Ex. 5: 3y x

Topic 4-3: One-to-One Functions Definition: A function f is said to be one-to-one

if for every value f(x) in the range of f there is exactly one corresponding x-value in the domain of f.

Ex. Are the following functions one-to-one over their domains?

2

3

3

2

2

2

2

f x x

g x x

h x x

H x x x

The Horizontal Line Test A function f is one-to-one if every horizontal line applied to the graph of f intersects f at most one. Are the graphs below representing the graphs of a function? Are they representing the graphs of a one-to-one function?

Any function which is not one-to-one over its entire domain can be amended to be one-to-one with appropriate restrictions on its domain.

Consider: 2

2g x x

Topic 4-4a: Inverse Functions Definition: An inverse function is a one-to-one

function such that if a one-to-one function f with domain A and range B exists, then the inverse function f

−1

(read as “f inverse”) has domain B and range A and the following property is satisfied:

1

f x y f y x

for all x in A and all y in B.

Inverse Operations Addition & Subtraction over ℝ Multiplication & Division over ℝ Odd Powers & Odd-Indexed Roots over ℝ Even Powers & Even-Indexed Roots over ℝ+

(or ℝ −

but not both)

Property of Inverse Functions For two functions f and g to be inverse functions of each other, the following conditions must be established:

1. f and g must be one-to-one functions over their given domains,

2. the domain of f must be the same as the range of g and the domain of g must be the same as the range of f, and

3. ( ) ( ) .f g x x g f x

Ex. 1 Determine if functions f and g are inverse functions of each other.

1

2 12

xf x x g x

Ex. 2 Determine if functions f and g are inverse functions of each other.

2

1 1, 0f x x g x x x

Finding an Inverse Function

To find the inverse function of a given function f:

1. Determine if f is one-to-one and its domain and range.

2. Apply one of the methods for finding the rule of

an inverse function. 3. Resolve any issues with the rule to ensure that

f −1

is one-to-one and has the appropriate domain and range.

To find the rule of an inverse function, you may apply one of two methods, each with its advantages and drawbacks. Verbal method: Write the steps of the function

that are applied to the variable. The inverse function’s rule will apply the inverse operations in the reverse order.

Symbolic method: Define the variable of the

function as y. Invert the roles of the independent variable (usually written as x) and y. Solve for y which will represent the rule of the inverse function.

Ex. 1 Find the inverse function of f.

3 4f x x


27

xf x


2

2 1, 0f x x x


2 5f x x

Topic 4-4b: Graphs of Inverse Functions

Since inverse functions effectively reverse the roles of the independent and dependent variables, the graph of an inverse function should satisfy the following pair of properties:

1. For every point (a, b) on the graph of f, the graph of f

−1 should include a point (b, a).

2. The graph of f −1

should demonstrate symmetry to f with respect to the line y = x.

Ex. 1 Find the graph of the inverse of the function graphed below. The find the domain and range of f

−1.

Ex. 2 Find the graph of the inverse of the function graphed below. The find the domain and range of f

−1.

Topic 4-5: Quadratic Functions Definition: A quadratic function is a function

which can be written as

2

, where 0.f x ax bx c a

Properties of Quadratic Functions 1. The graph of a quadratic function is a parabola. If a is positive, the parabola opens up. If a is negative, the parabola opens down.

1. Parabolas have a turning point called a vertex.

2. Parabolas have an axis of symmetry which is a vertical line passing through the vertex.

4. Quadratic functions have exactly one increasing interval and exactly one decreasing interval.

5. The vertex is an extremum. If the parabola opens up, the function value at the vertex is a minimum value. If it opens down, the function value is a maximum value.

The general form of a quadratic function provides clues about the vertex, axis of symmetry, and extreme value but requires investigation in order to find them.

Thus, a potentially more useful form of a quadratic function might provide direct information about these properties. Compare the graph of a basic squaring function.

2

f x x

To the graph of a transformed squaring function.

2

2 4f x x

Vertex Form of a Quadratic Function

2

f x a x h k

If a function is written in vertex form, how can we find the properties of parabola?

1. Direction of Opening: If a > 0, the parabola opens up.

If a < 0, the parabola opens down.

2. Vertex of the Parabola: Vertex is at (h, k).

3. Axis of Symmetry. The axis of symmetry is x = h.

4. Increasing/Decreasing:

If a > 0, f decreases over (−¥, h) and

increases over (h, ¥).

If a < 0. f increases over (−¥, h) and

decreases over (h, ¥).

5. Extreme function value: If the parabola opens up, f(h) = k is the

minimum value of f. If it opens down, f(h) = k is the maximum value of f.

By applying the completing the square technique, it is possible to rewrite a quadratic function into vertex form. Steps to put a quadratic function into vertex form:

1. Group the x² and x terms. 2. As relevant, factor a from the group. 3. Complete the square inside the parentheses. 4. Compensate the function outside the

parentheses. 5. Factor and simplify.

Ex. 1 Rewrite the function in vertex form by completing the square.

2

6 7f x x x

With the function in vertex form, it should be easy to answer the following questions about the graph of f: 1. Does the function open up or down? 2. What are the coordinates of the vertex? 3. What is the equation of the axis of symmetry? 4. Over what interval is the function increasing? decreasing? 5. Does the function have a minimum or maximum

value? What is the value and where does it occur?

Ex. 2 Rewrite the function in vertex form and then identify the properties associated with the function.

2

2 8 10f x x x

f opens ______

Vertex: ( ____ , ____ )

Axis of Symmetry: _____________

Increasing: _____________________

Decreasing: _____________________

y-intercept: _________

x-intercept(s): _____________________

Extremum has a __________ value of _________


2

3 6 2f x x x

f opens ______

Vertex: ( ____ , ____ )


Increasing: _____________________

Decreasing: _____________________

y-intercept: _________

x-intercept(s): _____________________ Extremum has a __________ value of _________


21

34 7f x x x

f opens ______

Vertex: ( ____ , ____ )


Increasing: _____________________

Decreasing: _____________________

f(x)-intercept: _________

x-intercept(s): _____________________

Extremum has a __________ value of _________

Finding a quadratic function from limited information. When given the vertex of a quadratic function, it is possible to uniquely find the rule of the function if you know only one other point on the parabola. Ex. 1 Find the function whose graph is a

parabola with vertex 3, 4 and passes

through the point 1, 8 .

Ex. 2 Find the function whose graph is a

parabola with vertex 1, 2 and passes

through the point 2, 4 .

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Topic 4-1: Algebra of Functions Do mains of Created