lesson 1.2, pg. 138 functions & graphs objectives: to identify relations and functions, evaluate...
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Lesson 1.2, pg. 138Functions & Graphs
Objectives: To identify relations and functions, evaluate functions, find the domain and range of
functions, determine whether a graph is a function, and graph a function.
Domain & Range
• A relation is a set of ordered pairs.• Domain: first components in the relation
(independent); x-values• Range: second components in the relation
(dependent, the value depends on what the domain value is); y-values
• Find the domain and range of the relation. {(5,12), (10, 4), (15, 6), (-2, 4), (2, 8 )}
FUNCTIONS
• Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.
Every “x” has only one “y”.
Mapping – illustrates how each member of the domain is paired with each member of the range (Note: List
domain and range values once each, in order.)
x y
0457
-912
Is this relation a function?
Draw a mapping for the following. (5, 1), (7, 2), (4, -9), (0, 2)
See Example 2, page 150.
Determine whether each relation is a function:A) {(1,2), (3,4), (5,6), (5,8)}
B) {(1,2), (3,4), (6,5), (8,5)}
Functions as EquationsDetermine whether the equation defines y as a function of x.
a) b)
1.Solve for y in terms of x.2.If two or more values of y can be obtained for a given x, the equation is not a function.
44 222 yxyx
Determine if the equation defines y as a function of x.
A) 2x + y = 6
B) x2 + y2 = 1
C) x2 + 2y = 10
Evaluating a Function
• Common notation: f(x) = function
• Evaluate the function at various values of x, represented as: f(a), f(b), etc.
• Example: f(x) = 3x – 7 f(2) = f(3 – x) =
If f(x) = x2 – 2x + 7, evaluate each of the following.
• a) f(-5) b) f(x + 4) c) f(-x)
See Example 4, page 143 for additional practice.
Determine if a relation is a function from the graph?
• Remember: to be a function, an x-value is assigned ONLY one y-value .
• On a graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line.
• THUS, the vertical line test! If a vertical line drawn on any part of your graph touches more than one point, it is NOT the graph of a function.
Graphs of Functions
Step 1: Graph the relation. (Use graphing calculator or pencil and paper.)
Step 2: Use the vertical line test to see if the relation is a function.
• Vertical line test – If any vertical line passes through more than one point of the graph, the relation is not a function.
Determine if the graph is a function.
a) b) y
x
5
5
-5-5
x
y
Here’s more practice.
c) d) y
x
y
x
Example
Analyze the graph.2( ) 3 4
a. Is this a function?
b. Find f(4)
c. Find f(1)
d. For what value of x is f(x)=-4
f x x x
x
y
(a)
(b)
(c)
(d)
Find f(7).
x
y
0
1
1
2
Can you identify domain & range from the graph?
• Look horizontally. What x-values are contained in the graph? That’s your domain!
• Look vertically. What y-values are contained in the graph? That’s your range!
• Write domain and range using interval or set-builder notation.
• See Example 8, page 148.
Domain: set of all values of xRange: set of all values of y
•Always write the domain and range in interval notation when reading the domain and range from a graph.•Use brackets [ or ] to show values that are included in the graph.•Use parentheses ( or ) to show values that are NOT included in the graph.
x
yIdentify the function's domain and range from the graph
Domain (-1,4]
Range [1,3)
Domain [3, )
Range [0, )
x
y
Example
Identify the Domain and Range from the graph.
x
y
Example
Identify the Domain and Range from the graph.
x
y
Example
Identify the Domain and Range from the graph.
x
y
(a)
(b)
(c)
(d)
Find the Domain and Range.
D:(- , ) R:(-5,7]
D:(-5, ) R: (- , )
D:(- , ) R: [-5, )
D:[- , ] R: [-5, ]
x
y
What is the difference in the two sets below, and when should we use each to describe the domain of a function?
{1,2,3,4} [1,4]
Finding intercepts:
• x-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero.
• y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero.
• Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!
We can identify x and y intercepts from a function's graph.
To find the x-intercepts, look for the points at which the graph
crosses the x axis. The y-intercepts are the points where the graph
crosses the y axis.
The zeros of a function, f, are the x values for which f(x)=0.
These are the x intercepts.
By definition of a function, for each value of x we can
have at most one value for y. What does this mean in terms
of intercepts? A function can have more than one x-intercept
but at most one y intercept.
Example
Find the x intercept(s). Find f(-4)
x
y
x
y
Example
Find the x and y intercepts. Find f(5).
Summary
• Domain = x values• Range = y values• Use the vertical line test to verify if a graph is
a function.• To evaluate means to substitute and simplify.• Intercepts – where function crosses the x-or y-
axis