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Functions, Function Notation, and Composition of Functions

Functions, Function Notation, and Composition of Functions. A Way to Describe a Relationship. 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 . .

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Functions, Function Notation, and Composition of Functions

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  1. Functions, Function Notation, and Composition of Functions

  2. A Way to Describe a Relationship 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.

  3. 4 Examples of Functions These are all functions because every x value has only one possible y value Every one of these functions is a relation.

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

  5. 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.

  6. 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.

  7. Function Notation: f(x) It does stand for “plug a value for x into a formula f” Does not stand for “f times 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: fof 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.

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

  9. A Justification for Function Notation 25 The new notation reduces the amount of writing needed to express this substitution and evaluation. For instance: Which do you prefer to write? 5 OR 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.

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

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

  12. 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

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

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