1.2 Combining Functions, Shifting, and Scaling Graphs

Operations on Functions

We can create new functions by combining two existing functions using addition, subtraction, multiplication, or division. These operations are fundamental in calculus and allow us to build more complex expressions from simpler ones.

• Addition:

• Subtraction:

• Multiplication:

• Division: , where

Domains: The domain of each new function is the intersection of the domains of and , except for division, where we must also exclude points where .

Example

Let and . Find , , , and , and their respective domains.

• Sum:

• Difference:

• Product:

• Quotient: ,

• Domain: For all except the quotient, the domain is all real numbers. For the quotient, .

Composition of Functions

The composition of two functions and , denoted , is defined as . The domain of consists of all in the domain of such that is in the domain of .

Example

Let and . Find and and their respective domains.

• ; domain:

• ; domain: all real numbers

Example

Let and . Find , , , and and their respective domains.

• ; domain: ,

• ; domain: ,

• ; domain: ,

• ; domain: ,

Additional info: The domains are found by excluding values that make any denominator zero or result in undefined expressions.

Transformations: Translation, Stretching/Shrinking, and Reflection

Transformations allow us to shift, stretch, shrink, or reflect the graph of a function. The table below summarizes common transformations and their effects:

Example

Use appropriate transformations to sketch the graph of the function .

• Start with , which is .

• Shift inside: shifts the graph horizontally.

• Subtracting $1 unit.

Additional info: When sketching, consider the domain where , i.e., or .