Division input/output tables

Learning Objectives Analyze an input/output table to determine the underlying division-based variable expression. Write the rule for a division input/output table using formal function notation, such as f(x) = P(x)/Q(x). Complete missing input or output values in a table governed by a rational function. Identify the domain restrictions of a function derived from a division input/output table. Connect the concept of an undefined output in a table to the vertical asymptote of the corresponding rational function's graph. Solve for the input value given a specific output, requiring algebraic manipulation of the rational expression. If a computer program's output is always a quotient related to its input, how can you reverse-engineer the code's core command? 💻 Th...

TermDefinitionExample Input (Independent Variable)The value that is provided to a function or rule. In a table, it is typically represented by 'x' or the first column/row.In the table for the rule y = 10/x, the values 1, 2, and 5 would be inputs. Output (Dependent Variable)The value that results from applying the function's rule to a given input. It is typically represented by 'y', 'f(x)', or the second column/row.For the rule y = 10/x, if the input is 2, the output is 5. Variable Expression (Function Rule)The algebraic formula that defines the mathematical relationship between the input and the output. In this context, it involves division.The expression (x + 3) / (x - 1) is a variable expression that can serve as a function rule. Rational FunctionA fun...

General Form of a Rational Function f(x) = \frac{P(x)}{Q(x)} This is the fundamental structure for any rule governing a division input/output table. P(x) and Q(x) are polynomial expressions. The key is to identify the specific expressions for P(x) and Q(x) based on the table's values. Inverse Variation f(x) = \frac{k}{x} A common type of division relationship where the output is inversely proportional to the input. The constant 'k' is the constant of variation. To find 'k' from a table, multiply any input 'x' by its corresponding output 'f(x)'. If the product is constant across all pairs, the rule is inverse variation. Domain Restriction Rule Set Q(x) = 0 and solve for x. To find the values that are NOT in the domain of a...