Functions and Limits

In this lecture we shall discuss:

What a function is
Notation for functions
Domain of a function
Range of a function

Function
If a quantity y depends on another quantify x in such a way that each value of x determines exactly one value of y, we say that y is a function of x

y=4x+1 defines y as a function of x because each value assigned to x determines exactly one value of y.

x	Value of y = 4x + 1
2	9
1	5
0	1
-1/4	0
√3	4√3 + 1

In the following example y is not a function of x, as each value assigned to x determines two values of y.
y = ± √x
if x = 4
y = ±√4
y = 2 and y = -2

If we use the letter f to denote a function, then the equation

y = f(x)

y is a function of x

Although f is the symbol most commonly used to denote a function, any symbol can be used. Thus

y = F(x)
y = g(x)
y = h(x)

Example
φ(x) = 1/(x³ - 1)
Then
φ( ³√7) = 1/(x³ = 1/( ³√7)³ - 1) = 1/(7 - 1) = 1/6
φ(1) = 1/[(1)³ - 1] = 1/0 Undefined

Example
F(x) = 2x² - 1
F(d) = 2(d)² - 1
F(t - 1) = 2(t - 1)² - 1
= 2(t² -2t + 1) -1
= 2t² - 4t + 1

g(c) = c² - 4c

g(x) = x² - 4x

Range of a Function
For every value given to the independent variable from the domain in a function, we get a corresponding y value.

The set of all such y values is called the range of the function

Example
h(x) = 1/[(x - 1)/(x - 3)]
The Domain is
(-∞, 1) ∪ (1, 3) ∪ (3, +∞)

Example
h(x) = (x² - 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2) = (x + 2) x ≠ 2

f(x) = x²
Rewrite as:
y = x²

Natural Domain

If a function is defined by a formula and there is no domain explicitly stated, then it is understood that the domain consists of all real numbers for which the formula makes sense, and the function has a real value. This is called the natural domain of the function.

Example
y = (x + 1)/(x - 1) - The natural domain is all reals except 1
Solve for y
x = (y + 1)/(y - 1) - Range is all reals except 1
Piecewise-defined Functions

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