Functions, Function Notation, and Composition of Functions

A Way to Describe a Relationship RelationA relationship between sets of information. Typically between inputs and outputs. FunctionA relation such that there is no more than one output for each input We have worked with many mathematical objects. For instance: equations, rules, formulas, tables, graphs, etc. In mathematics, similar things can also be described by the following vocabulary.

4 Examples of Functions XY XY These are all functions because every x value has only one possible y value Every one of these functions is a relation.

3 Examples of Non-Functions XY Every one of these non-functions is a relation. Not a function since x=1 can be either y=10 or y=-3 Not a function since x=-4 can be either y=7 or y=1 Not a function since multiple x values have multiple y values

The Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

The Vertical Line Test If a vertical line intersects a curve more than once, it is not a function. Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

Function Notation: f(x) Equations that are functions are typically written in a different form than “ y =.” Below is an example of function notation: The equation above is read: f of x equals the square root of x. The first letter, in this case f, is the name of the function machine and the value inside the parentheses is the input. The expression to the right of the equal sign shows what the machine does to the input. Does not stand for “f times x” It does stand for “plug a value for x into a formula f”

Example If g(x) = 2x + 3, find g(5). You want x=5 since g(x) was changed to g(5) When evaluating, do not write g(x)! You wanted to find g(5). So the complete final answer includes g(5) not g(x)

A Justification for Function Notation A function is similar to a factory machine. For the machine below, when 25 is the input (raw product) to the machine below, the output (finished product) is 5. OR 25 5 The new notation reduces the amount of writing needed to express this substitution and evaluation. For instance: Which do you prefer to write?

Solving v Evaluating Substitute and Evaluate The input (or x) is 3. Solve for x The output is -5. No equal signEqual sign

Substituting a function or it’s value into another function. Composition of Functions f g First (inside parentheses always first) Second OR

Example 1 Let and. Find: This is an equivalent way to write it (The book does not use this notation): Substitute x=1 into g(x) first Substitute the result into f(x) last

Example 2 Let and. Find: Substitute x into f(x) first Substitute the result into g(x) last