Here’s what you need to know, generally, about functions.
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[Note: “iff” means “if and only if.”]
Definition
A function is a relationship between two sets of numbers, one containing inputs and the other for outputs; these sets are called "the domain" and "the range," respectively. A function can can be understood as an input/output “machine” that takes a number in and returns a corresponding number out, such that no input is associated with more than one output. Normally, the input is called x and the output is called y. The function itself is named with a single letter, like f, in which case the output for general input x can be written “f(x),” pronounced “f of x.”
y = f(x)
y and f(x) are interchangeable.
Function values
The “value of a function” is an output value (y value).
Operations
The essential operation with functions is substitution.
“g(n)” means substitute n for x in function g.
Composition of Functions
Composite functions are “nested” functions. “f[g(x)]” means function g is nested inside function f.
For example: To find f[g(2)], first find g(2) and then substitute that value into f.
Zeros of a function
Values of x (input values) that make y (output values) equal zero.
Zeros are found at x-intercepts.
The “value of a function” is an output value (y value).
Operations
The essential operation with functions is substitution.
“g(n)” means substitute n for x in function g.
Composition of Functions
Composite functions are “nested” functions. “f[g(x)]” means function g is nested inside function f.
For example: To find f[g(2)], first find g(2) and then substitute that value into f.
Zeros of a function
Values of x (input values) that make y (output values) equal zero.
Zeros are found at x-intercepts.
When f(x) = 0, solutions are called “roots.”
Solutions iff roots iff zeros iff x-intercepts (“roots,” “zeros,” “solutions,” “x-intercepts” are essentially synonymous).
Intercepts of functions:
To find intercepts, let the other variable’s value be zero
Solutions iff roots iff zeros iff x-intercepts (“roots,” “zeros,” “solutions,” “x-intercepts” are essentially synonymous).
Intercepts of functions:
To find intercepts, let the other variable’s value be zero
For example: For the y-intercept, let x = 0).
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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.
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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.
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Copyright © 2006-present: Christopher R. Borland. All rights reserved.
