Section 1.4 (part 2)- Compositions of Functions

This section includes forming and solving the compositions of functions,  finding the domain of a composite function, identifying composite functions, and looking into problems in which 
f · g=g · f.


Forming and Solving the Compositions of Functions
The composition of functions is a way of combining two functions. The composition of the function f with g is:


(f · g)(x)=f(g(x))

Example: a) Find (f · g)(x) for f(x)=x-3 and g(x)=x² + 4. Evaluate (f · g)(x) when x=3.

(f · g)(x)=f(g(x))

=f(x²+4)

=(x²+4)-3

=x²+1

(f · g)(3)=(3)²+1=10

b) Find (g · f)(x) for f(x)=x-3 and g(x)=x² + 4. Evaluate (g · f)(x) when x=3.

(g · f)(x)=g(f(x))

=g(x-3)

=(x-3)²+4

=(x²-6x+9)+4

=x²-6x+13

(g · f)(3)=(3)²-6(3)+13=4

*Note that (f · g)(x)≠(g · f)(x)

Finding the Domain of a Composite Function

The domain of f · g is the set of all x in he domain of g such that g(x) is in the domain of f. 

  In the chart above, x is the domain of g, and g(x) is the domain of f.

To determine the domain of a composite function such as (f · g)(x), you must restrict the outputs of g(x) so that they are in the domain of f. The domain of g(x) must also be taken into consideration, because 

only the outputs of g will be put into f.

Example: a) Find the domain of the composition (f · g)(x) for f(x)=1/x and g(x)=x-3.

(f · g)(x)=f(g(x))

=f(x-3)

=1/(x-3)

There is no restriction on the domain of g, so the outputs of g can be any real number. The domain of f 

is restricted to all real numbers besides 0. This means that g(x)=x-3≠0, so x≠3. 

The domain of (f · g)(x) is all real numbers except x=3.

                b) Find the domain of the composition of (f · g)(x) for f(x)=x²+2 and 

(f · g)(x)=f(g(x))

=4-x²+2

=2-x²

It may appear that the domain of this function is all real numbers, but this is not the case. The function g  has a restriction on its domain, and therefore there is a restriction on the composition of the functions.

The domain of g is . So, the domain of (f · g)(x) is  also.

Identifying Composite Functions

It is also important to be able to identify the two functions that make up a composite function. To do this, look for an "inner" and an "outer" function. In a composite function (f · g)(x), the outer function will be the f(x) function and the inner function is the g(x) function.

Example: Express the function  as a composition of two functions. 

inner function: g(x)=3x+1

outer function: f(x)=1/x²

this allows you to write this as a composite of two functions:

Looking into Problems in which (f · g)=(g · f)

There are some instances when (f · g)(x)=(g · f)(x); the composition of these functions is the same in both cases.

Example: Find (f · g)(x) and (g · f)(x) given that f(x)=2x+3 and g(x)=0.5(x-3).

(f · g)(x)=f(g(x))

=f(0.5(x-3))

=2(0.5(x-3))+3

=x-3+3

=x

(g · f)(x)=g(f(x))

=g(2x+3)

=0.5((2x+3)-3)

=0.5(2x)

=x

Note that no matter what the value of x is, (f · g)(x) will always equal (g · f)(x), because the composition of the functions is the same.

Chapter 1.4 Combinations of Functions

This chapter covers arithmetic combinations of functions.  This includes the sum, difference, product, and quotient of functions. 


*Note: in order for two functions to be combined they must have an overlapping domain

Sum of Two Functions 

The sum of two functions, f(x) and g(x), can be expressed as follows:

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

Ex. Find the sum of the two functions, f(x)= 2x+1 and g(x)= 9x+11

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

(f+g)(x)= 2x+1+9x+11

(f+g)(x)=11x+12

When graphing the sum of two functions the domain remains the same as the original functions but the y values are that of the two functions added together for each value for x.

Example: the combination of the y values on the blue graph and the y values of the gray graph, at any value of x, yield the y values for red graph. This concept can be applied to all graphs for arithmetic combinations of functions.  

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&docid=PvlUamPmfM90uM&tbnid=rDmM6IhEmlkzzM:&ved=0CAQQjB0&url=http%3A%2F%2Fvhcc2.vhcc.edu%2Fpc1fall9%2Fproperties_of_functions%2Fcombining_graphs_of_functions.htm&ei=_ciwUobgAcPsyQGf8ICIBw&bvm=bv.58187178,d.aWc&psig=AFQjCNEOVUrOec4l_FbA-vKGDAlkmjW7Tw&ust=1387403866403779

Difference of Two Functions

The difference of two functions, f(x) and g(x), can be expressed as follows: 

(f-g)(x)= f(x)-g(x)

Ex. Find the difference of two functions f(x)=13x-5 and g(x)=7x+6

(f-g)(x)= f(x)-g(x)

(f-g)(x)= 13x-5-(7x+6)   

(f-g)(x)= 13x-5-7x-6

(f-g)(x)= 4x-11

*Always remember to include parenthesis and distribute the negative to all terms when subtracting the second function!!!

When graphing the difference of two functions the domain remains the same as the original functions but the y values are that of the second function subtracted from the first function for any value of x.

Product of Two Functions

The product of two functions, f(x) and g(x), can be expressed as follows: 

(fg)(x)= f(x)g(x)

Ex. Find the product of the two functions f(x)=27x + 1 and g(x)= x+2

(fg)(x)= f(x)g(x)

(fg)(x)= (27x+ 1)(x+2)

When graphing the product of two functions the domain remains the same as the original functions but the y values are that of the y values for each function multiplied together for any value of x.  

Quotient of Two Functions

The quotient of two functions, f(x) and g(x), can be expressed as follows: 

(f/g)(x)= f(x)/g(x)

Ex. Find the quotient of the two functions and

(f/g)(x)= f(x)/g(x)

Domain: [0, ∞)

*Note: Often when providing the equation for the quotient of two functions it is often asked that the domain be specified for the function to provide all real solutions 

When graphing the quotient of two functions the domain remains the same as the original functions but the y values are that of the y values for each function divided by one another for any value of x. 

Chapter 1.3- Shifting, Reflecting, and Stretching Graphs

Chapter 1.3

Chapter 1.3 explains how vertical and horizontal shifting, reflecting, and stretching graphs is possible.

Parent functions are basic functions that can be graphed very easily. Some common parent functions include identity functions, absolute value functions, and square root functions.

Parent functions can have transformations occur to them, and in result produce daughter functions that are similar to the parent functions except that they are transformed or altered in specific way. These transformations include:

Vertical and Horizontal Shifting

Vertical shifting to a parent function can cause the function to either move up or down the vertical axis depending on if the number added to the function is positive or negative. Positive number = graph moves upward that many units. Negative number = graph moves down that many units on the y axis

f(x)= x² is the parent function (red graph)

And f(x) = x²+3 is the daughter function (blue graph) that is shifted 3 units up on the y axis since 3 is a positive number.

 

 If the parent function again is f(x)= x²

And f(x) =x² – 2 a transformed version, the original function on a graph will be shifted two units downward on the y axis

 

Horizontal shifting is when the parent function is transformed by shifting to either the left or the right along the x axis. This can occur when a number is either added or subtracted directly from x. When a positive number is added to x, the function shifts to the left. When a negative number is added to x, the function is shifted to the right.

When the original function is

f(x)=x (red graph) and then it is transformed by adding 2, f(x)=(x=2) (blue graph)  the new function is shifted to the left by two units on the x axis.  

 

 

When the same parent function, f(x)=x, is altered by subtracting 4, f(x)=x-4, the original function is shifted to the right 4 units on the x axis.





Overview:

1. Vertical shift c units upward:                    h(x) = f(x) + c

2. Vertical shift c units downward:               h(x) = f(x) - c

3. Horizontal shift c units to the right:          h(x) = f(x - c)

4. Horizontal shift c units to left:                  h(x) = f(x + c)



Reflecting Graphs

A Reflection can be considered a mirror image of the parent function  and is reflected either in the x-axis or in the y-axis.

Reflections in the coordinate axes of the graph of y = f(x) are represented as follows:
1. reflection in the x-axis:                  h(x)= -f(x)
2. reflection in the y-axis:                  h(x)= f(-x)

If the parent function is

 (red)

Then the reflection in the x-axis of this function,  (blue)  are both graphed like:

 

 

If the reflection in the y-axis were found from the same parent function,  , then the new equation would be  f(x)= and the graph would look like:

***Side Note***

Rigid transformations include horizontal shifts, vertical shifts, and reflections. They are rigid transformations because the basic shape of the graph goes unchanged, only the position on the graph is altered in the xy plane.

Nonrigid transformations on the other hand do change the basic shape of a function (distortion). These type of transformations include vertical and horizontal stretches which will be discussed below.

 

Stretching Graphs

 

 A vertical stretch is an alteration or distortion to a parent function by making the function skinner and lengthier.

A vertical compress is an alteration or distortion to a parent function by making the function wider.

 

A nonrigid transformation of the graph of y = f(x) is represented by y = cf(x)

               a vertical stretch occurs if c > 1

               a vertical compress occurs if 0 < c < 1

 

If the parent function is: (red graph)

 A vertical stretch can be:   f ( x ) = 4 |x| (blue graph) , so the graph is:

 

If the parent function is:

A vertical shrink can be:  f ( x ) = (1/2) |x| , so the graph looks like:

 

For horizontal stretches, the nonrigid transformation of the graph of y = f(x) is represented by            y = f(cx).
             a horizontal compress is when c > 1
             a horizontal stretch is when 0 > c >1

If the parent function is:

 The horizontal condense can be:

If there is the same parent function, and the function is horizontally stretched, then a new function could be:

Chapter 1.2

Chapter 1.2 focuses on graphs of functions. The topics that we’ll discuss are Graphs of Functions, Increasing and Decreasing Functions, Relative Minimum/Maximum, Step and Piecewise-Defined Functions, and Even/Odd Functions.

Graphs of Functions

A graph of a function is all the combined points of (x, f(x)) plotted on a graph. X is the input of the function and f(x) is the output.

Consider the following function:

 We can find out the domain and range of a function by looking at its graph.

In this example, the domain and range are both  

, as any value of x corresponds to a value of f(x) and vice-versa.

This isn't always the case, however. Consider this next function:

Also, look at its graph.

From this graph, you can see that the domain excludes zero, and that the range excludes all negative values and zero.

Increasing and Decreasing Functions

A piece of a graph of a function can be described as increasing, decreasing, or remaining constant.

A piece of a graph is considered increasing if, as x increases, f(x) increases.
 If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).

A piece of a graph is considered decreasing if, as x increases, f(x) decreases.
  If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).

A piece of a graph is considered remaining constant if, as x increases, f(x) doesn't change.

 If, for any x1 and x2 in the interval, x1 < x2 implies f(x1) = f(x2).

Relative Minimum/Maximum

A relative maximum is a point in a graph within two x values that has the highest value for f(x). Said in another way, if you make two imaginary, invisible, vertical lines in a graph, the highest point in between those is the relative maximum. Similarly, a relative minimum is a point in a graph within two x values that has the lowest value for f(x).

Formerly speaking, a function value f(a) is a relative minimum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) <= f(x).
A function value f(a) is a relative maximum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies f(a) >= f(x).

Step and Piecewise-Defined Functions

A step function is a noncontinuous function that remains constant for a defined amount of x values, then jumps suddenly to a new y value. Step functions often resemble staircases.

(graph from mathwords.com)

Piecewise-defined functions are functions that have multiple components, depending on the value of x.

Consider the following piecewise-defined function.

Now look at its graph.

Even and Odd Functions

A function is considered even if the graph of it has symmetry with respect to the y-axis (f(-x) = f(x)).

A function is considered odd if the graph of it has symmetry with respect to the origin (f(-x) = -f(x)).

Consider the following function.

f(-x) = (-x)^2

f(-x) = x^2

f(-x) = f(x)

Therefore, this is an even function.

You can also tell this is an even function by looking at its graph.

As you can see, it is reflected across the y axis, so it is an even function.