1. Let's start by understanding what a function is. A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. 2. The general form of a function is written as $f(x)$, where $x$ is the input and $f(x)$ is the output. 3. To formulate a function, you need to identify the rule that assigns each input $x$ to an output $f(x)$. 4. For example, if the rule is "multiply the input by 2 and then add 3," the function can be written as: $$f(x) = 2x + 3$$ 5. Important rules to remember: - Each input $x$ must have exactly one output $f(x)$. - Functions can be represented by equations, tables, graphs, or words. 6. Let's formulate a function from a word problem: "A taxi charges a base fare of 4 plus 2 per mile." - Let $x$ be the number of miles. - The function is: $$f(x) = 4 + 2x$$ 7. To evaluate the function for $x=5$ miles: $$f(5) = 4 + 2 \times 5 = 4 + 10 = 14$$ 8. This means the taxi fare for 5 miles is 14. 9. Another example: If a function doubles the input and subtracts 1, write it as: $$f(x) = 2x - 1$$ 10. To check if a relation is a function, ensure no input corresponds to more than one output. Formulating functions involves identifying the input-output rule and expressing it mathematically using $f(x)$ notation.