1. An Introduction to Functions
Digital Lesson
An Introduction to Functions

2. A relation is a rule of correspondence that relates two sets.
For instance, the formula I = 500r describes a relation between the amount of interest I earned in one year and the interest rate r.
In mathematics, relations are represented by sets of ordered pairs (x, y) .
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3. The set A is called the domain of the function.
A function from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B.
The set A is called the domain of the function.
The set B is called the range of the function.
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4. Function Set B Set A y x Domain Range
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5. Example: Determine whether the relation represents y as a function of x.
b) {(-1, 1), (-1, -1), (0, 3), (2, 4)}
Not a Function
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6. Functions represented by equations are often named using a letter such as f or g.
The symbol f (x), read as “the value of f at x” or simply as “f of x”, is the element in the range of f that corresponds with the domain element x. That is,
y = f (x)
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7. The domain elements, x, can be thought of as the inputs and the range elements, f (x), can be thought of as the outputs.
Function
Input
Output f x
f (x)
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8. Example: Let f (x) = x2 – 3x – 1. Find f (–2).
To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.
Example: Let f (x) = x2 – 3x – 1. Find f (–2).
f (x) = x2 – 3x – 1
f (–2) = (–2)2 – 3(–2) – 1 Substitute –2 for x.
f (–2) = – Simplify.
f (–2) = The value of f at –2 is 9.
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9. f (x + 2) = 4(x + 2) – (x + 2)2 Substitute x + 2 for x.
Example: Let f (x) = 4x – x2. Find f (x + 2).
f (x) = 4x – x2
f (x + 2) = 4(x + 2) – (x + 2) Substitute x + 2 for x.
f (x + 2) = 4x + 8 – (x2 +4x + 4) Expand (x + 2)2
f (x + 2) = 4x + 8 – x2 – 4x – Distribute –1.
f (x + 2) = 4 – x The value of f at x + 2 is – x2.
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10. Example: Find the domain of the function f (x) = 3x +5
The domain of a function f is the set of all real numbers for which the function makes sense.
Example: Find the domain of the function
f (x) = 3x +5
Domain: All real numbers
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11. Example: Find the domain of the function
The function is defined only for x-values for which x – 3  0. Solving the inequality yields
x – 3  0
x  3
Domain: {x| x  3}
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12. Example: Find the domain of the function
The x values for which the function is undefined are excluded from the domain. The function is undefined when x2 – 1 = 0.
x2 – 1 = 0
(x + 1)(x – 1) = 0
x =  1
Domain: {x| x   1}
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