1.2 Graphs of FUNctions

1.2 Graphs of FUNctions

Domain and Range of a Function
The Domain of a function is essentially how far the x- values extend in a graph. The Range is a representation of how far the y- values extend.


As an example, in the graph below, y = x² + 2.
          - The domain of this graph would be (-∞, ∞) because any given input would be able to produce a valid output. The graph extends infinitely because it is a parabola
          - The range would be a bit more limited. Because every given input is added to two, the output can at minimum two. As long as it is more than two, any y value is achievable, so the range for this graph is [2, ∞)

In another case, even the domain of a graph can be limited. In the graph below, y = √(x+3)

          - The domain of this graph would have to be at least -3. This is because anything below that would put a negative under the radical sign, which would bring imaginary numbers into the picture, and we want to avoid that. The inequality 0  ≤ x+3 can be used to determine this. When solved, it is apparent that x can equal anything greater than or equal to three, thus the domain is [3, ∞]
          - The range of this graph could be anything about zero. Because of the square root, we know that the output will always be positive, so y will equal anything greater than or equal to zero. Thus, the domain  is [0, ∞]

* Graphs generated via online graphing calculator at http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html


A function can be tested for graphically by using the Vertical Line Test

A vertical line is drawn through a graph. The line should only go through the graph once to be a considered a function. This would imply that at most one y value corresponds with each x value, which is the definition of a function. If the vertical line intersects with the graph more than once, it means that an x value corresponds with more than y value, making it not a function

 

Increasing and Decreasing Functions

As in the textbook:

For any x₁ and x₂ in the interval,


- A function increasing when x₁ < x₂ implies that
 
f(x₁ ) < f(x₂)
- A function decreasing when x₁ < x₂ implies that
f(x₁ ) < f(x₂)
- A function constant when f(x₁ ) = f(x₂)
   

 For example:

To determine the intervals upon which the graph above is increasing, decreasing, or held constant, the direction in which the graph is travelling at different points is to be taken into account.

- From -5 and on towards the left the graph is parallel to the x axis, thus held constant, so the graph is constant at (-∞, -5)

- From -2 to -6, the graph is moving upward towards the left, thus decreasing, so the graph is decreasing at (-2, -6)

-From -5 and on, the graph is moving upward towards the right, thus increasing, so the graph is increasing at (-2, ∞)

Even and Odd Functions

Even Functions

                Functions are even when they are symmetric to the y-axis

                                In an even function,  f(-x) = f(x), as shown in the graph below

Odd Functions

                Functions are odd when they are symmetric to the origin

                                In an odd function, f(-x) = -f(x), as shown in the graph below

Testing for even or odd functions

To algebraically determine whether a function is even odd, or neither, substitute in a –x for x, and simplify.

 

For example:

h(x) = x³  + 5x

h(-x) = ( -x)³– 5x

        = -x³-5x

        = -h(x)

h(-x) = -h(x)

ODD

f(x) = x² + 1

f(-x) = (-x)² +1

        = x² + 1

f(-x) = f(x)

EVEN

f(x) = x² + 2x – 3

f(-x) = (-x)² – 2x – 3

f(-x) ≠ f(x)

f(-x) ≠ -f(x)

NEITHER

 Relative Maximum and Minimum Functions

A function of f(a) is called a relative minimum of f if there is an interval  (x_1, x₂) that contains a such that

 x₁ < x < x₂ implies f(a) ≤ f(x)

A function of f(a) is called a relative maximum of f if there is an interval (x_1, x₂) that contains a such that

 x₁ < x < x₂ implies f(a) ≥ f(x)