KEY FEATURES OF FUNCTIONS
Following are the key features of functions.
1. Domain and Range
2. x-intercept and y-intercept
3. Positive and Negative intervals
4. Intervals of increasing, decreasing and constant behavior
5. Parent Functions
6. Maxima and Minima
Domain and Range
The domain is the set of all possible inputs or x-values. To find the domain of a function, we have to look at the x-axis of the graph.
Determining Domain :
1. Start at the origin.
2. Move along the x-axis until you find the lowest possible x-value. This is your lower bound.
3. Return to the origin.
4. Move along the x-axis until you find your highest possible x-value. This is your upper bound.
The range is the set of all possible outputs or y-values. To find the range of the graph, we have to look at the y-axis of the graph.
Determining Range :
For the range, do the same thing but move along the y-axis.
x-intercepts and y-intercepts
x-intercept :
1. This is where the graph crosses the x-axis.
2. To find it algebraically, set y = 0.
3. Have many names :
- x-intercept
y-intercept :
1. This is where the graph crosses the y-axis.
2. To find it algebraically, set x = 0.
Positive and Negative Intervals
Positive Interval :
In the diagram above, the graph of the function is above the x-axis in the following intervals.
(-3, -1) and (2, 4)
More precisely, y is positive when x ∈ (-3, -1) and (2, 4).
So, the positive intervals for the above graph are
Negative Interval :
In the diagram above, the graph of the function is below the x-axis in the following intervals.
(-∞, -3), (-1, 2) and (4, + ∞)
More precisely, y is negative when x ∈ (-∞, -3), (-1, 2) and (4, + ∞ ) .
So, the negative intervals for the above graph are
(-∞, -3), (-1, 2) and (4, +∞)
Types of Function Behavior
There are three types of function behavior :
1. Increasing
2. Decreasing
3. Constant
When determining the type of behavior, we always have to move from left to right on the graph.
1. Increasing :
- When x increases, y will also increase
- Direct variation
2. Decreasing :
- When x increases, y will decrease
- Inverse variation
3. Constant :
- When x increases, y will stay the same
Identifying Intervals of Behavior
We use interval notation to represent the behavior of the function.
The interval measures x-values. The type of behavior describes y-values.
In the diagram above,
* the graph is increasing in the intervals :
(a, b) and (c, d)
* the graph is decreasing in the interval :
Parent Functions and Their Graphs
The most basic for a type of function.
Determines the general shape of the graph (the end behavior).
Baby Functions
Look and behave similarly to their parent functions.
To get a 'baby' function, add, subtract, multiply, and/or divide parent function by constants.
Function Name :
Absolute Value
Parent Function :
f(x) = |x|
Baby Function :
f(x) = |x - 1|
Identifying Parent Functions
From equations, identify the most important operation :
- Special Operations (Absolute Value)
- Division by x
- Highest Exponent (this includes square roots and cube roots)
1. f(x) = x2 + 5x + 6
2. f(x) = 3 / (x + 2)
3. f(x) = 3|x| + 5
Maximum (Maxima) and Minimum (Minima) Points
Maximum Point (Maxima) :
Peaks (or hills) are the maximum points.
Minimum Point (Minima) :
Valleys are the minimum points.
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