Introduction to Function Notation

• 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 $\quad$ Outputs

  $-1 \quad \longrightarrow \quad -3$

  $\ \ 0 \quad \longrightarrow \quad -1$

  $\ \ 1 \quad \longrightarrow \quad \ \ 1$

  $\ \ 2 \quad \longrightarrow \quad \ \ 3$

• 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.

• The entire expression $f(x)$ represents the final output, which is equivalent to $y$ on a coordinate graph.

• You may also see arrow notation, such as $f: x \to 3x + 2$, where the colon and arrow show the journey from input to rule.

• To evaluate a function, you substitute a specific numerical value in place of $x$.

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 $3$) with $f(x) = 3$ (which means set the expression equal to $3$).

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 $x = k$ must be excluded because division by zero is impossible.

• Similarly, for a square root function like $f(x) = \sqrt{x-k}$, the domain must be $x \geq k$ to avoid square roots of negative numbers.

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 $x \neq 4$.