Section 1.4- Composition Combinations 

COMPOSITION COMBINATIONS

The composition combination of functions is similar to that of arithmetic combinations, most commonly used for operations such as sale tax. Involving at least two functions, composition combinations involve the input of one function into another function. Unlike arithmetic combinations, which involve using arithmetic operations in order to combine functions, composition combinations use a function as an input for another function, replacing variables.

The main type of composition combination is as follows:

Composition Function

 Here is an example of the composition function:

Example Function

Now combine f(x) and g(x) by replacing the variable in f(x) with g(x)

Now simplify and find the answer

If you were to graph this function, it would look like this

 Domain and Range of Composite Functions

Composite combinations involve the combining of two functions, which means that when graphing the end simplified equation the other functions must also be possible. This means that the range and domain of all equations can only be graphed and technically the only correct answers. This occurs when impossible solutions are possible in later equations. This is explained in the following picture.

All impossible answers are also in impossible in the end equation no matter what transformation occurs. Here's an example:

Composition Domain and Range

 The domain of f(x) is all real numbers, while the domain of g(x) is greater than or equal to 1 or -1. Now lets finish the equation.

The end composite equation's domain is all real numbers, yet because the g(x) equation has a domain greater than or equal to 1 or -1, the entire composition equation must have a domain greater than or equal to one.

1.4 Arithmetic Combinations

ARITHMETIC COMBINATIONS

The arithmetic combination of functions is similar to the combination of real numbers. It is commonly seen in the real world to calculate the number of supplies a company has, or various other types of payment methods.

There are 4 types of Arithmetic Combinations, and they are as follows:

Sum

 

 

Difference

 

 

 Product 

 

Quotient

 

How do we use these functions in a problem you may ask?

 

Sum:

 

Ex. Findfor the functions:

 

       and

 

 

Make sure to combine all like terms and distribute negative signs appropriately!!

If you were to graph this equation it would look like this:

 

 



 

 Difference:

 

Ex. Findfor the functions:

 

 and

 

 

Make sure to combine all like terms and distribute negative signs appropriately!!

If you were to graph this equation it would look like this:

 

 

Product:

 

 

Ex. Find for the functions:

 

 

and

 

 

Tip: When distributing, use the FOIL method (first, outside, inside, last)

If you were to graph this equation it would look like this:

 



 

 



Quotient

 

 Ex. Find  for the functions:

 

and

 

 

 

 The second part of the answer for quotient arithmetic combination problems is the domain.

 

To find the domain:

1. Set the denominator of the quotient equal to zero

2. Solve for 'x'

 

Ex.

 

 **This will often be a follow-up question when you are finding the quotient**

 If you were to graph this equation it would look like this:

 

 

 

 

 



Section 1.3- Shifting, Reflecting, and Stretching Graphs

Common Parent Functions and Their Graphs

Parent Function-the simplest form of a family of functions 

a. Absolute Value Function

b. Logarithmic Function

c. Square Root Function 

d. Identity Function 

e. Quadratic Function 

f. Cubic Function 

g. Constant Function 

In the case of this graph, c=3 

Often, functions are simple transformations of the parent functions

transformations-the transformations discussed in this section are:

~vertical and horizontal shifts

~reflections across the x-axis and the y-axis

~vertical and horizontal stretching and compressing

Vertical and Horizontal Shifts

Vertical Shifts move a graph up or down by adding or subtracting a positive, real number from f(x)

Vertical shift c units upward: 

Vertical shift c units downward:

Parent Function

Vertical Shift up 2 units (c=2)

Vertical Shift down 3 units (c=3)

The parent function is shown being shifted 2 units up and 3 units down. In the two transformed graphs, a number (c) is being added or subtracted from f(x).

Horizontal Shift
Horizontal shifts move a graph left or right by adding or subtracting a positive, real number from the input of f(x).

Horizontal shift c units to the right:

Horizontal shift c units to the left:

Parent Function

Horizontal Shift right 1 unit

Horizontal Shift left 5 units

The parent function is shown being shifted 5 units left and 1 unit right. In the two transformed graphs, a number (c) is being added or subtracted from the input of f(x).

Reflecting Across the x-axis and y-axis

Reflection in the x-axis: 

Reflection in the y-axis:

 Parent Function

Reflection in the y-axis

Reflection in the x-axis

The parent function is shown being reflected in the x-axis and reflected in the y-axis. In the two transformed graphs, the opposite of the output or the opposite of the input is being taken.

If a graph looks the same when reflected over the x-axis and the y-axis, it is an odd function.

Example: 

 

The reflection of the function over the x-axis and the y-axis yields the same graph, meaning the original function is odd.

If a graph looks the same as the original when reflected over the y-axis, the function is even.

Example:

The reflection of the function over the y-axis is the same as the graph of the original function, meaning the original function is symmetrical about the y-axis and even.

Vertical and Horizontal Stretching and Compressing

Vertically Stretching and Compressing

f(x) = cf(x) 

Where c is the factor by which the graph is vertically stretched or compressed

Vertical stretch if c (1,∞)

Vertical compression if c (0,1)

Parent Function

Verticle Stretch

Verticle Compression

f(x) = 0.5x²

The parent function is shown being compressed vertically be a factor of 0.5 and stretched vertically by a factor of 2. In the two transformed graphs, a number (c) is being multiplied by the output of the function f(x).

Horizontal Stretching and Compressing

f(x) = f(cx)

Where c is the factor by which the graph is horizontally stretched or compressed

Horizontal compresses if c (1,∞)

Horizontally stretches if c (0,1) 

Original Function

f(x) = x³-2x+2

Horizontal Compression

f(x) = (2)x³-2(2)x+2

f(x) = 2x³-4x+2

Horizontal Stretch

f(x) = (0.5)x³-2(0.5)x+2

f(x) = 0.5x³-x+2

The original function is shown being stretched horizontally be a factor of 2 and compressed horizontally by a factor of 0.5. In the two transformed graphs, a number (c) is being multiplied by the input of the function f(x).

✮When applying multiple transformations to a graph, abide by the order of operations✮