GradeGen.AI

Function Machines: Inputs and Outputs

Every time you use a vending machine, you put money in, press a button, and get a snack out.
A mathematical function works the same way: it is a rule that assigns exactly one output to each input.
The simplest way to show this is a function machine (also called a number machine), which consists of an input, one or more operation boxes, and an output.
For example, a function machine for the rule "multiply by $2$ , then add $3$ " looks like this: $\text{Input} \to [\times 2] \to [+ 3] \to \text{Output}$

Operations must be applied in the exact sequence shown from left to right, following standard order of operations.
To find an input when you only know the output, you must work backwards using an inverse operation.
This means reversing the order of the machine and doing the opposite operations in reverse sequence.
Working backwards through the machine above to find the input would look like this: $\text{Input} \leftarrow [\div 2] \leftarrow [- 3] \leftarrow \text{Output}$

Mapping Diagrams

A mapping is a process identifying how elements from the domain are paired with the range.
In a mapping diagram, the left oval contains the inputs and the right oval contains the outputs, with arrows connecting them.
For example, a mapping diagram for $f(x) = 2x - 1$ with the domain $\{-1, 0, 1, 2\}$ looks like this:

Inputs $\quad$
For a mapping to be a true function, every input must have exactly one output.
A one-to-one function gives a unique output for every input, while a many-to-one function has multiple inputs sharing the same output (like $f(x) = x^2$ , where $-2$ and $2$ both map to $4$ ).
A mapping where one input maps to multiple outputs is a one-to-many relation, but this is NOT a function.
You can test this on a graph: if a vertical line crosses the graph more than once, it is a one-to-many mapping.

Formal Function Notation

Once you understand function machines, you can write them using formal algebraic notation.
Functions are typically written as $f(x)$ , but you may also see $g(x)$ or $h(x)$ .
The letter $f$ is the name of the function, and the $x$ inside the brackets represents the input.

Calculate $f(-2)$ given $f(x) = 3x^2 - 5x + 2$ .

Step 1: Substitute the value $-2$ into the expression, using brackets for negative numbers.

$f(-2) = 3(-2)^2 - 5(-2) + 2$

Step 2: Calculate the powers and multiplications first.

$f(-2) = 3(4) + 10 + 2$

Step 3: Complete the addition for the final output.

$f(-2) = 24$

Solving Equations: $f(x) = k$

Evaluating a function means substituting a value in, but solving an equation means finding the input that produces a specific output.
To solve $f(x) = k$ , you must set the algebraic expression equal to the number $k$ and solve for $x$ .
Do not confuse $f(3)$ (which means substitute ) with (which means set the expression equal to ).

Solve $f(x) = 17$ given $f(x) = 2x + 7$ .

Step 1: Set the function's expression equal to the given output.

$2x + 7 = 17$

Step 2: Rearrange to isolate the $x$ term.

$2x = 10$

Step 3: Solve for $x$ .

$x = 5$

Domain, Range, and Excluded Values

The domain is the set of all possible input values ( $x$ ) for which the function is defined.
The range is the set of all possible output values ( $f(x)$ or $y$ ) produced by the function.
Some functions have restricted domains because certain inputs would make the math undefined.
For example, in a fraction like $f(x) = \frac{1}{x-k}$ , the value must be excluded because division by zero is impossible.

Find the value of $x$ that must be excluded from the domain of $f(x) = \frac{4}{2x - 8}$ .

Step 1: Identify the operation that would make the function undefined (the denominator cannot be zero).

$2x - 8 = 0$

Step 2: Solve for $x$ to find the excluded value.

$2x = 8$
$x = 4$

Step 3: State the restricted domain.

The domain is $x \neq 4$ .

Students often apply inverse operations in the original order when working backwards through a function machine; you must reverse the sequence of operations as well as the signs.

When substituting negative numbers into a function, always use brackets in your calculator, especially with powers, to avoid incorrect signs (e.g., $(-3)^2 = 9$ , not $-3^2 = -9$ ).

In solving questions like $f(x) = 17$ , examiners expect you to state $x = 5$ explicitly on the answer line; do not leave your answer embedded in the calculation like $2(5) + 7 = 17$ .

When asked to state a domain or range, use $x$ for the domain and $f(x)$ for the range, as using $x$ for the range is frequently penalised in mark schemes.

A mathematical rule or machine that assigns exactly one output to each input.

The initial value put into a function or number machine.

The final result after all mathematical operations have been applied to the input.

A diagram representing a mathematical rule where operations are applied to an input in a specific sequence.

A mathematical operation that undoes the effect of another, such as addition undoing subtraction.

A visual process identifying how elements from the domain are paired with the range.

A mapping where each input has a unique output, which means it has a valid inverse.

A valid function where multiple inputs share the same output, such as squaring negative and positive numbers.

A mapping where a single input maps to multiple outputs; this is not considered a true mathematical function.

The process of substituting a specific numerical value into a function to find the output.

The complete set of all possible input values for which a function is mathematically defined.

The complete set of all possible output values produced by a function.