Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 7 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Find a function value from a point on the graph
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Function notation names an input, while the requested function value is the output paired with it. On the graph, we’ll begin at the marked x-coordinate, move vertically to the plotted …
Preview problemFind domain and range of a finite relation
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Separate the coordinates by role before collecting any values: first coordinates are inputs and second coordinates are outputs. We’ll form one set from each position and remove repeated entries, since …
Preview problemFind domain and range from a continuous graph description
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Think of the graph casting a shadow onto each axis. Its horizontal extent gives the domain, and its vertical extent gives the range; continuity tells us whether every value between …
Preview problemDecide whether a real-world mapping can be modeled as a function
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
First define the mapping exactly as the context states it, including what counts as one input and one output. Then ask whether a single allowed input could receive two different …
Preview problemMatch a linear function rule to a graph description
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Start by matching the rule’s structure to a familiar graph family. For a first-degree rule in slope-intercept form, the x-coefficient controls the rate of change and the constant locates the …
Preview problemIdentify multiple inputs with the same output
Understand functions as mappings from inputs to exactly one output; connect f(x) to graph y=f(x).
Translate each function statement into an input-to-output mapping, keeping the value inside parentheses separate from the value after the equals sign. We’ll compare the outputs first, locate any repeated value, …
Preview problemEvaluate a function at a given input by substitution
Use function notation, evaluate functions on their domains, and interpret notation in context.
Function evaluation means feeding the given input into the rule, not reporting the input itself. We’ll replace every occurrence of the variable with the supplied number, use parentheses to preserve …
Preview problemFind a function value from a table using function notation
Use function notation, evaluate functions on their domains, and interpret notation in context.
The number inside the function notation tells us which input row to find. We’ll locate that value in the table’s x-column and read straight across the same row to its …
Preview problemFind a function value from a graph
Use function notation, evaluate functions on their domains, and interpret notation in context.
On a graph of y equals f of x, the number inside the parentheses is a horizontal coordinate, while the function value is vertical. We’ll start at the requested x-value, …
Preview problemInterpret function notation in context
Use function notation, evaluate functions on their domains, and interpret notation in context.
Replace the abstract notation with the quantities named in the context. The number inside the parentheses is an input with its own units, and the value of the function is …
Preview problemFind the input value(s) that produce a given output from a function rule
Use function notation, evaluate functions on their domains, and interpret notation in context.
This question runs function evaluation backward: the output is known, and the input is unknown. We’ll set the function expression equal to the target output, undo the operations in reverse …
Preview problemEvaluate a function at an algebraic input
Use function notation, evaluate functions on their domains, and interpret notation in context.
An algebraic expression can be a single function input, so keep the entire replacement grouped at first. We’ll insert that grouped expression wherever the original input variable appears, preserve the …
Preview problemEvaluate a piecewise function
Use function notation, evaluate functions on their domains, and interpret notation in context.
For a piecewise function, branch conditions come before arithmetic. We’ll compare the given input with each condition, identify the one interval that contains it, and use only that branch’s formula. …
Preview problemCheck domain before evaluating a function
Use function notation, evaluate functions on their domains, and interpret notation in context.
Domain comes before evaluation, because a formula can produce written arithmetic even when the real-valued function forbids the input. We’ll derive the square-root restriction from the entire radicand, test the …
Preview problemInterpret input and output units in function notation
Use function notation, evaluate functions on their domains, and interpret notation in context.
Use the function’s definition to attach a quantity and unit to each side of the mapping. The value inside parentheses inherits the input variable’s unit, while the function value inherits …
Preview problemCompare two function values
Use function notation, evaluate functions on their domains, and interpret notation in context.
A comparison of function values must be based on outputs, not just on the size of the inputs. We’ll substitute each input into the same rule in separate calculations, preserve …
Preview problemInterpret function notation
Use function notation, evaluate functions on their domains, and interpret notation in context.
Read function notation as a machine name followed by its supplied input. Because a particular input is already inside the parentheses, the task is to determine the single output assigned …
Preview problemIdentify the domain of a sequence
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
A sequence is a function whose inputs are term numbers, so its domain comes from the indexing rather than from an interval on the real line. We’ll identify the first …
Preview problemWrite a sequence as ordered pairs using the given term numbering
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
Represent each sequence term as an input-output pair: the term number comes first, and its value comes second. We’ll honor the stated starting index, assign consecutive indices to the listed …
Preview problemGraph a sequence
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
On a sequence graph, the horizontal coordinate is the term number and the vertical coordinate is that term’s value. We’ll plot exactly the listed ordered pairs and read the domain …
Preview problemDecide whether a sequence graph represents a function
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
Discrete points can represent a function without forming a continuous curve or a straight line. We’ll inspect each plotted input and count how many points occur directly above it; one …
Preview problemFind a sequence term from an explicit rule, table, or recursive rule
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
The requested sequence subscript is the function input, so first confirm that term number belongs to the stated integer domain. Then substitute it for n in the explicit rule and …
Preview problemInterpret a recursive sequence definition as a function
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
A recursive sequence needs both a starting input-output pair and a rule that advances from one integer input to the next. We’ll use the initial subscript to establish the domain’s …
Preview problemDecide whether explicit and recursive rules define the same sequence
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
Different-looking formulas define the same sequence only if they agree at every allowed index. We’ll first compare their domains and initial outputs, then identify the term-to-term change in each representation …
Preview problemModel a finite list as a sequence function
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
A sequence may be finite as long as its outputs are ordered and indexed. We’ll assign consecutive integer term numbers to the listed values in their given order, state the …
Preview problemDetermine the range of a sequence from a table, graph of points, or listed terms
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
The range comes from the sequence outputs, not from the term-number inputs. We’ll identify the output column, collect every value it actually contains, and remove any repeats because a range …
Preview problemInterpret a sequence rule in context
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
Interpret the sequence index as the discrete contextual input before reading the formula’s parameters. Evaluating at the first allowed index reveals the initial amount, while increasing the index by one …
Preview problemDetect an invalid sequence-function representation
Recognize sequences, including recursive sequences, as functions whose domains are subsets of integers.
Read every ordered pair as a term-number input followed by its sequence output. We’ll group pairs with the same first coordinate and look for one input assigned different second coordinates; …
Preview problemInterpret a y-intercept in context
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
A vertical intercept is the graph’s output when the input is zero. We’ll read the marked point in input-output order, attach the axis units, and translate it into function notation. …
Preview problemInterpret the x-intercept or zero of a function in context
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
A horizontal intercept is an input where the function’s output reaches zero. We’ll read the marked point with the axis units, place its input and output correctly into function notation, …
Preview problemIdentify where a function is increasing or decreasing
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Read the graph from left to right and locate every input where its direction changes. We’ll describe whether outputs rise, fall, or stay level on each interval separated by those …
Preview problemInterpret interval behavior in context
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Begin by naming the input interval with its units and the output quantity whose behavior is being described. Increasing means later inputs in that interval produce greater outputs. We’ll compare …
Preview problemFind maximum and minimum values from a table
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
Maximum and minimum refer to function outputs, so scan the entire output column before looking back at inputs. We’ll identify the greatest and least displayed values, then trace each one …
Preview problemInterpret an extremum point in context
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
An extremum point carries two different facts: when the extreme occurs and what the extreme output is. We’ll use the context to assign quantities and units to the first and …
Preview problemIdentify symmetry from a graph description
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
A graph is symmetric when one transformation leaves its shape unchanged. We’ll locate the curve’s center or turning point, then compare points the same horizontal distance on either side of …
Preview problemDescribe end behavior
Interpret and sketch key graph/table features in context: intercepts, intervals, maxima/minima, symmetry, end behavior, and periodicity.
End behavior is really two separate questions: what happens far to the right and what happens far to the left. We’ll follow the graph in each direction and translate that …
Preview problem