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2.1 – What Is a Function?

Objectives

  • Distinguish between relations and functions.
  • Calculate domain and range of functions algebraically.
  • Identify domain and range of functions graphically.

 

Key Terms

  • Dependent Variable – A variable in a function whose value is determined by the value of the independent variable.
  • Domain – The set of all possible input (x-values) of the independent variable.
  • Function – A special type of relationship in algebra where two quantities are related to each other so that one quantity depends on (or is a function of) the other. For each input number, there is exactly one output number.
  • Independent Variable – The input variable in a function (x-value). Its value determines the value of the dependent variable. Graphed on the x-axis.
  • Interval Notation – A shorthand way of writing intervals using parentheses and brackets.
  • Range – The set of all possible output (y-values) of the dependent variable. The difference between the maximum value and the minimum value of a data set, written as range = maximum − minimum.
  • Relation – A set of ordered pairs [(x,y) pairs]. x is the domain / input / independent variable. y is the range / output / dependent variable. If each element in the domain is paired with just one element in the range, the relation is also a function. A relation is sometimes called a mapping.

 

Notes

Naming Functions
  • Naming Functions: Standard Form: y = f(x)
    • How to write it: “y is a function of x” is written as y = f(x).
    • How to say it: the algebraic expression says “y equals f of x.”
    • What it means:
      • y depends on x
      • The quantity y is a function of the quantity x
      • x is input (independent) and y is output (dependent)
      • Each value of x will result in exactly one value of y.
      • Function Name: f is the function name
      • This function is the rule for converting x into y
    • Graphed: x are the x-values (horizontal) and y are the y-values (vertical)

  • Example (with other variables)
    • Write it: “C is a function of t” is written algebraically as “C = g(t)”
    • Say it: “C equals g of t”
    • Means:
      • C depends on t
      • The quantity C is a function of the quantity t
      • t is the input (independent) and C is the output (dependent)
      • Each value of t will result in exactly one value of C.
    • Function Name: g is the function name
      • This function is the rule for converting t into C
    • Graphed: t are the x-values and C are the y-values

  • Example in Words
    • The amount of time it takes to finish a race might be a function of the distance of the race
Domain
  • Domains (Graphed)
    • To find the domain, look at the drawing on the graph.
    • Notice how far LEFT and RIGHT the graph can go.
    • All the x-values on the graph
      • Domain Limits – Sometimes, the x-values have limits.
        • Explicit (direct) limits are given to you:
          • You were given that x must be greater than or equal to zero in this example.
        • Inferred (indirect) limits are those in which you must think about algebraic rules you already know: 
          • Ask yourself: Can x by any number?  Are there any limits to having a variable in the denominator?
          • In this example, x cannot be 4 because, if you substitute 4 for x, then you get 4 – 4 = 0 and you cannot have 0 in the denominator of a fraction, ever!
          • Answer: x ≠ 4
          • You should write this domain in Interval Notation like this:
            • See more about Interval Notation below
        • Another inferred (indirect) example of limits with a radical
          • Ask yourself: Are there any limits to having a variable inside a square root (radical)?
          • In this example, x can be equal to -2 (because the square root of zero is zero); however, x cannot be less than -2 because you cannot have a negative value inside a square root (it would be an imaginary number).
          • Answer: x ≥ -2
          • You should write this domain in Interview Notation like this:
        • Graph

Alg2A - 2.01 - Graph of Radical

Range
  • Range (Graphed)
    • To find the range, look at the drawing on the graph.  Notice how far UP and DOWN the graph can go.
    • All the y-values on the graph

  • Graph Example 1

Alg2A - Domain Range Ex 01

  • For this graph:
    • Domain: All real numbers
      • … because the graph goes to the left infinitely and to the right infinitely.
      • Also written in Interval Notation like this:
    • Range: y ≥ 0;  y is greater than or equal to zero
      • … because there are no y-values below zero on the graph.  All of the y-values on this graph start at zero and go up.
      • Also written in Interval Notation like this:

  • Graph Example 2

Alg2A - 2.01 - All Real x and y

  • For this graph:
    • Domain: All real numbers
      • … because the graph goes to the left infinitely and to the right infinitely
      • Also written in Interval Notation like this:
    • Range: All real numbers
      • …because the graph goes to the left infinitely and to the right infinitely
      • Also written in Interval Notation like this:

  • Graph Example 3

Alg2A - 2.01 - Horizontal D Rjpg

  • Domain: All real numbers because it goes left and right infinitely
  • Range: is the set {3} because it does not go above or below 3

  • Graph Example 4: Is the following graph of a relation also a function?

Alg2A - 2.01 - Not a Function

  • This graph shows a relation that is NOT a function
    • …because the value x = 3 is assigned to more than one y-value
      • On this graph, we can see points (3, -3), (3, -2), (3, -1), (3, 0), (3, 1), (3, 2), (3, 3), etc.
    • Remember: functions have only 1 y-value for each x-value
      • For this relation, when x = 3, y is { … -3, -2, -1, 0, 1, 2, 3 … }
Domain and Range
  • Think of domain and range like a machine.
    • You put in a domain and you get out a range.
  • Ex. Like cooking: you put in the ingredients (domain) and you get out the cake (range)

 

Alg2A - Domain Range Img


  • Real World Example (see the Baby growth chart below)
    • What are your x (input values)?  What are your y (output values)?
    • Remember: x is the domain, y is the range

Alg2A - Table Set

Evaluating a Function
  • Example f(x) = x2 + 2x.
    • What is the value of this function when x = 5? This is another way of asking, what is the f(5)?
    • To find f(5), just substitute 5 in place of each x in the function and calculate the answer.
      • f(5) = 52 + 2(5)
      • f(5) = 25 + 10
      • f(5) = 35
    • To find f(a), substitute a into every input (x-value)
      • f(a) = a2 + 2(a)
    • To find f(a + 3), substitute (a + 3) into every input (x-value)
      • f(a + 3) = (a + 3)2 + 2(a + 3)
Interval Notation
Alg2A - 2.01 - Symbols Interval Notation

 

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