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( 37) = 1/(x3 = 1/( 37)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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