Input-Output View Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Think of f(x)=3x−7f(x) = 3x - 7 as a machine. Describe the sequence of operations applied to input xx, then evaluate f(5)f(5) and find the input that gives output 1414.

Solution

  1. 1
    Machine description: take input xx → multiply by 33 → subtract 77 → output.
  2. 2
    Evaluate: f(5)=3(5)−7=15−7=8f(5) = 3(5)-7 = 15-7 = 8.
  3. 3
    Find input for output 1414: solve 3x−7=14⇒3x=21⇒x=73x-7=14 \Rightarrow 3x=21 \Rightarrow x=7.

Answer

f(5)=8f(5)=8; input x=7x=7 gives output 1414
The input-output view treats a function as a process or machine. This perspective makes it natural to evaluate forward (given input, find output) and backward (given output, find input), building intuition for inverse operations.

About Input-Output View

The input-output view of a function treats it as a black box: put in a value (input), get out a uniquely determined value (output), without worrying about the internal mechanism.

Learn more about Input-Output View →

More Input-Output View Examples

Example 2 medium

A function machine applies two operations in sequence: first square the input, then add [formula]. W

Example 3 easy

A vending machine charges [formula]1.50[formula]C(n)[formula]n[formula]C(4)[formula][formula].

Example 4 medium

Two function machines are connected: machine [formula] doubles the input, machine [formula] subtract

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