Compositions of Functions

One way of combining two different functions is to form the composition of one with the other. The domain of this is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function f with g is:

For example:

The domain is then

When determining the domain of a composite function, you need to restrict the outputs of g so that they are in the domain for f. For example, to find the domain of f composed with g of x given that f(x)=1/x and g(x)=x+1, note the outputs of g. They can be any real number, but the domain of f is restricted to all real numbers except 0. Therefore, the outputs of g must also be restricted to all real numbers except 0. This means that in g(x)=x+1≠ 0, or x ≠ -1. So, the domain of f composed with g of x is all real numbers except x=-1.

Next is to find the domain of the domain of f composed with g of x for the functions:

It may look like the domain of the composition is the set of all real numbers. Although, this is incorrect because the domain of g is:

Thus, the domain of f composed with g of x is also:

Finally, we can identify a composite function by expressing a function as a composition of two functions. We may not know f(x) or g(x) at first:

Arithmetic Combinations of Functions

Arithmetic Combinations of Functions

Functions can be combined arithmetically by the sum, difference, product, or quotient.

Sum:                           

Difference:             

Product:                        

Quotient:                        

The domain of an arithmetic combination of functions f and g includes all real numbers that are in both of the domains of f and g. In the case above, all of the domains are all real numbers except for the quotient f(x)/g(x). The denominator cannot be zero when combining functions with a quotient. In this quotient, g(x) cannot equal zero. Since g(x) cannot equal zero, zero acts as a vertical asymptote when graphing the combined function.

Example: If and, you can form the sum, difference, product, and quotient of f and g.

Sum:

Difference:

Product:

Quotient:

The denominator cannot equal zero. In this case, the denominator is (x - 1). Set the denominator to zero and isolate x to find the vertical asymptote. 

Add one to both sides

One is the vertical asymptote

Transformations of Functions

All types of functions have a parent function, the simplest type of function in a family of functions. Some common ones are:  , , ,, 

Functions can be manipulated by a vertical shift, horizontal shift, vertical stretch, horizontal shrink, or a reflection 

Horizontal and Vertical shifts: depending on where x is manipulated will cause a horizontal or vertical shift. If the x is added or subtracted outside of the parenthesizes, then the graph will shift up or down. If the x is added or subtracted inside of the parenthesizes, then the graph will shift left or right.

Vertical shift up: 

Vertical shift down:  

Horizontal shift to the right: 

Horizontal shift to the left: 

Vertical Stretch and Horizontal Shrink: a horizontal shrink and a vertical stretch affect the width of a function. In a vertical stretch or shrink, the function is multiplied by c outside of the parenthesizes. If c<1, then it will be a vertical shrink. If c>1, then it will be a vertical stretch. In a horizontal stretch or shrink, the function is multiplied by c inside of the parenthesizes. If c>1, then it will be a horizontal shrink. If c<1, then it will be a horizontal stretch.

Vertical stretch: 

Horizontal shrink: 

There are ways to tell if the graph has been affected vertically or horizontally. In a vertical stretch or shrink, there is no change in the x-intercepts. In a horizontal stretch of shrink, there is no change in the relative minimums or maximums or y-intercepts.

Reflections: a reflection mirrors the function over either the x-axis or y-axis. When –c is multiplied outside of the parenthesizes, the function reflects over the x-axis. When –c is multiplied inside of the parenthesizes, the function reflects over the y-axis.

Reflection over x-axis: 

Reflection over y-axis: 

Reflections also bring to light the idea of even and odd functions. If f(-x) = f(x), then the function is even. If f(-x)= -f(x), then the function is odd.  However, if f(-x) does not equal either, then the function neither odd nor even.