Class Notes: Evaluating Functions Using a Graph

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Objective

Understand how to evaluate a function using its graph, including finding specific function values, intercepts, the domain & range, and determining if a relation is a function.

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Part A: Evaluating Function Values from the Graph

1. What is the value of f(-4)?

• Approach: Locate the x-value of -4 on the graph. Find the corresponding y-value (vertical coordinate) at that point.
• Result:
  • ( f(-4) = 5 ) (When ( x = -4 ), ( y = 5 )).

2. What is the value of f(4)?

• Approach: Locate the x-value of 4. Follow the graph to the corresponding y-value.
• Result:
  • ( f(4) = 3 ) (When ( x = 4 ), ( y = 3 )).

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Part B: Finding x-value When y is Given

1. At what point is ( f(x) = -7 )?

• Given: y = -7
• Approach: Find where the graph crosses y = -7; identify corresponding x-value.
• Result:
  • At ( x = 5 ), ( f(x) = -7 )
  • Ordered pair: (5, -7)

2. At what point is ( f(x) = 6 )?

• Given: y = 6
• Approach: Horizontal line y = 6 meets graph at two points.
• Result:
  • At ( x = -5 ) and ( x = 7 ), ( f(x) = 6 )
  • Ordered pairs: (-5, 6) and (7, 6)

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Part C: Inverse Functions

What is the value of ( f^{-1}(7) )?

• Given: y = 7
• Approach: Find x when y = 7.
• Result:
  • ( x = -6 )
  • ( f^{-1}(7) = -6 )

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Part D: Intercepts

1. X-Intercepts

• Definition: Where the graph crosses the x-axis (( y = 0 ))
• Identify:
  • Examples:
    • (-1, 0)
    • (3, 0)
  • Some intercepts are between points, may need estimation or line equations for exact values.

2. Y-Intercept

• Definition: Where the graph crosses the y-axis (( x = 0 ))
• Identify:
  • (0, 4)

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Part E: Domain and Range

1. Domain (Possible x-values)

• From graph:
  • Lowest ( x ): -6
  • Highest ( x ): 7
  • Domain: [ -6, 7 ] (brackets mean endpoints included)

2. Range (Possible y-values)

• From graph:
  • Lowest ( y ): -7
  • Highest ( y ): 7
  • Range: [ -7, 7 ]

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Part F: Is ( f(x) ) a Function?

1. Vertical Line Test

• Rule: A graph represents a function if any vertical line crosses it at most once.
• Result:
  • Passes the test; f(x) is a function.

2. Inverse Function

• Horizontal Line Test:
  • Fails; some horizontal lines cross the graph more than once.
  • Therefore, the inverse is not a function.

3. Function Type

• Not one-to-one: Some y-values have multiple x-values (e.g., y = 6 yields ( x = -5 ) and ( x = 7 )).

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Recap

• Evaluating using graph: Find the given x-value to get y, or given y-value to get x.
• Intercepts: X- and Y-intercepts are where graph crosses axes.
• Domain/Range: Determined by extremal x and y on the graph.
• Function Test: Vertical line test for function, horizontal for inverse.
• One-to-one: f(x) is a function, but not one-to-one since some y-values map to more than one x.

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Summary:
This lesson explains how to use a graph to evaluate functions and their inverses, determine intercepts, find domain/range, and check if a graph represents a function or a one-to-one function.