Input-Output View Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Input-Output View.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Like a vending machine: put in selection (input), get out snack (output).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Functions process inputs into outputs—focus on transformation.

Common stuck point: The same function can be viewed as mapping, process, or formula.

Sense of Study hint: Try describing the function as a sequence of steps: 'take the input, then do ___, then do ___, and the result is the output.'

Worked Examples

Example 1

easy
Think of f(x) = 3x - 7 as a machine. Describe the sequence of operations applied to input x, then evaluate f(5) and find the input that gives output 14.

Solution

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

Answer

f(5)=8; input x=7 gives output 14
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.

Example 2

medium
A function machine applies two operations in sequence: first square the input, then add 3. Write the function formula f(x), fill in a table for x \in \{-2,-1,0,1,2\}, and identify any symmetry.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A vending machine charges \1.50 per item. Write a function C(n) for the cost of n items, and find C(4) and the number of items for budget \9.

Example 2

medium
Two function machines are connected: machine A doubles the input, machine B subtracts 1. Find the combined output for input x=5, and write the combined formula.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition