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How Do You Find the Output of a Function?

Published in Functions 3 mins read

The output of a function depends on how the function is represented, but generally, you find the output by plugging in a specific input value into the function's rule or observing the corresponding y-value on its graph.

Here's a breakdown of how to find the output depending on the representation of the function:

1. Function Expressed as an Equation:

  • Explanation: This is the most common way functions are represented. The equation defines the relationship between the input (usually x) and the output (usually y or f(x)).
  • Process: To find the output for a given input, substitute the input value for the variable (usually x) in the equation and simplify. The result is the output value.
  • Example:
    • Function: f(x) = 2x + 3
    • Input: x = 4
    • Output: f(4) = 2(4) + 3 = 8 + 3 = 11. The output is 11.

2. Function Expressed as a Graph:

  • Explanation: The graph of a function visually represents the relationship between inputs (x-axis) and outputs (y-axis).
  • Process:
    1. Locate the input value on the x-axis.
    2. Draw a vertical line from that point to the graph of the function.
    3. From the point where the vertical line intersects the graph, draw a horizontal line to the y-axis.
    4. The value where the horizontal line intersects the y-axis is the output value corresponding to the given input.
  • Key Idea: Each point on the graph (x, y) represents an input-output pair, where 'x' is the input and 'y' is the output.

3. Function Expressed as a Table:

  • Explanation: A table lists input-output pairs explicitly.

  • Process: Find the input value in the input column. The corresponding value in the output column is the output for that input.

  • Example:

    Input (x) Output (f(x))
    1 5
    2 7
    3 9
    • If the input is 2, the output is 7.

4. Function Expressed as a Set of Ordered Pairs:

  • Explanation: The function is defined by a set of pairs in the form (input, output).
  • Process: Search the set for the ordered pair whose first element (the input) matches the given input. The second element of that pair is the output.
  • Example: {(1, 4), (2, 5), (3, 6)}
    • If the input is 2, the output is 5.

In summary, finding the output of a function involves identifying the rule that connects inputs to outputs, and then applying that rule to a specific input value.

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