Domains and Ranges from Graphs

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Domains and Ranges from Graphs

• Procedure for Finding Domains and Ranges from
  Graphs
• If a point is on the graph of a relation, then
  the input value is in the domain and the output
  value is in the range.
• When no output corresponds to a particular input
  value, the input value is not in the domain.
• When no point has a particular output value, the
  output value is not in the range.

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Example 1Finding the Domain and Range 1

• Use the graph to find the domain and range of the
  function.

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Example 2 Finding the Domain and Range 2

• For each graph, find the domain and range. Then
  tell whether the relation is a function or not.
• a. b.

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For each graph, find the domain and range. Then
tell whether the relation is a function or not.

• c. d.

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The y-intercept

• The point where the graph of a function
  intersects the vertical axis is called the
  y-intercept.
• In the function shown in below, the y-intercept
  is the point (0, 10), indicated by the red point.
  A mapping diagram shows the correspondence.

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Zeros of Functions

• The zeros of a function are the x-values that
  produce an output of 0 in the function, that is,
  when f(x) 0.
• An x-intercept is a point on the graph of a
  function where the graph intersects the x-axis.
  At these points, every point has a y-value of 0.
• Because each point on the x-axis has a y-value of
  0 and has a real number x-value, the real zeros
  of a function are the x-coordinates of the
  x-intercepts.
• If a graph does not touch the x-axis, the
  function has no real zeros and there are no
  real-number solutions to the equation f(x) 0.

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Example 3 Intercepts of a Function

• The graph of the function f is shown below.

For each expression, interpret the symbols, find
the value(s), and label the value(s) on the
graph. a.f(0) b. All x-values where f(x) 0.
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Example 4 Finding Positive and Negative Function
Values

• Find the intervals where the following function
  is positive and where it is negative.

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Indicate positive and negative values of the
function on the graph.
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Maxima and Minima

• Many situations are best explained by determining
  when output values reach a temporary top or
  bottom. Such points are called local maxima or
  local minima.
• A local maximum can be thought of as the top of a
  hill.
• A local minimum can be thought of as the bottom
  of a valley.

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Increasing and Decreasing Function Values

• Reading a graph from left to right, a function is
  
• increasing when the output values are rising and
• decreasing when the output values are falling.

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Example 5 Increasing, Decreasing, Maximum and
Minimum

• For the given function
• Approximate the local maximum and local minimum
  function values, stating which x-values produce
  each.
• Estimate the intervals where the function is
  increasing and the intervals for which it is
  decreasing.

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Find the Intervals of Increase and Decrease,
Maximum and Minimum Values
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• Procedure for Analyzing Functions
• A function can be described from its graph by
  finding
• the domain and range
• the y-intercept, if one exists
• the x-intercepts, if any exist
• the x- values that produce positive or negative
  function values
• the local maximum and minimum function values and
  the corresponding x-values. If applicable,
  identify which of these is an absolute maximum or
  minimum.
• the x- values where the function is increasing or
  decreasing

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Example 6 Analyzing a Function

• Analyze the following function, which
  approximates the low temperatures in C at a city
  in northern North Dakota during 2002. The input
  values are the days of the year and the output
  values are the low temperatures.
• Source United States Historical Climatology
  Network

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Note Even though the domain (days of the year)
is discrete, the number of days can be thought of
as continuous.