Intermediate Algebra
Introduction to Functions
Rules that turn every input into exactly one output: find them, write them, graph them, and run them backward.
THE SYLLABUS dev · all lessons unlocked
I
0 / 6
▶
Finding the Rule
Find the one rule that sends every input to its output, for numbers, days and words.
Free
▶
Word Functions
Rules that take in words: find the rule, predict what comes out, and work backward to what went in.
Open
▶
Time Functions
Rules that move a clock: find how many hours they add or take away, then run them forward and backward.
Open
▶
Repeating Outputs
Many inputs, one output: divisibility, remainders and rounding, found, used and run backward.
Open
▶
Deducing the Rule
Two rules can fit the same pairs. Each new pair is a clue that rules one of them out.
Open
▶
Level Check
Ten questions across Level 1: rules for numbers, words and times, run forward and backward.
Open
II
0 / 5
▶
Numeric Rules
Name a function and write its rule: f(x) = x + 3. Then use the notation forward, backward, and to build pairs.
Open
▶
Writing Rules
Turn a rule in words into a formula and back. The order of the steps decides where the parentheses go.
Open
▶
Conditional Rules
One function, two formulas: a condition decides which formula each input gets.
Open
▶
Deducing Numeric Rules
Each new pair is a new point, and it can rule out the formula that fit before. Absolute value and piecewise rules appear.
Open
▶
Level Check
Ten questions across Level 2: notation, formulas, two-step rules and piecewise rules, picked and typed.
Open
III
0 / 8
▶
Graphing Linear Functions
Inputs go across, outputs go up: plot a function's pairs and a linear rule draws a straight line.
Open
▶
Graphing Squared Functions
Squared rules bend: plot their pairs and the points curve into a U or an arch.
Open
▶
Graphing Absolute Value
Absolute value strips the sign: its graph is two straight pieces meeting at a sharp corner.
Open
▶
Graphing Without an Equation
A function needs no single formula to be graphed: plot it input by input, using whichever piece applies.
Open
▶
Interpreting Graphed Functions
Read a function off its graph: its highest and lowest values, where they happen, and where it crosses zero.
Open
▶
Intercepts and Intervals
Where a graph crosses the axes, and the stretches where it stays positive or negative.
Open
▶
Increasing and Decreasing
Where a graph climbs, where it falls, where it stays level, and the points where it turns.
Open
▶
Level Check
Fifteen questions across Level 3: plot rules that switch, read values off graphs, and a look back at Levels 1 and 2.
Open
IV
0 / 5
▶
Function Requirements
A rule is a function only if every input leads to exactly one output: one road, never a fork.
Open
▶
Vertical Line Test
If a vertical line meets a graph more than once, one input has two outputs, so the graph is not a function.
Open
▶
Valid Functions
A rule is a function only if every input gets exactly one output, every time; a graph is one only if no vertical line meets it twice.
Open
▶
Valid Intervals
A curve that fails the vertical line test can still be a function over an interval: the full stretch of x where it passes.
Open
▶
Level Check
Fifteen questions across Level 4: full intervals, counted outputs, the line that proves a curve fails, and a look back at Level 1.
Open
V
0 / 5
▶
Discrete Functions
Some functions only have outputs at separate inputs, like whole years; their graphs are points, and you can read them off one by one.
Open
▶
Depreciation Functions
A function can predict a price from another input, like age: read up from the age to find the value, or across from the value to find the age.
Open
▶
Multi-Part Functions
A multi-part function follows a different rule in each band of inputs: read it band by band, where it is flat, where it climbs, and where it climbs fastest.
Open
▶
Periodic Functions
A periodic function repeats the same pattern again and again: read its lowest and highest values, when it peaks, and when it rises or falls.
Open
▶
Level Check
Fifteen questions across Level 5: read rises, losses, swings and streaks off real-world graphs, then place and read points on a grid.
Open
VI
0 / 6
▶
Piecewise Functions
A piecewise function follows different rules for different inputs, written with a curly brace: plot it piece by piece and match it to its graph.
Open
▶
Discontinuities
Where a piecewise function jumps, an empty circle marks where a piece stops just short and a filled circle marks the function's real value: read the circles to match rules to graphs.
Open
▶
Plotting Points
Read graphs that jump, forward and backward: the filled circle holds the value, an empty circle never counts, and many inputs can share one output.
Open
▶
Plotting Piecewise Functions
Go from a graph to its formula: read each piece, then set each condition by its circles - a filled end means or-equal, an empty end means strictly.
Open
▶
Building Piecewise Functions
Build a piecewise function from scratch so its graph passes through every given point: set the conditions, then the pieces.
Open
▶
Level Check
Fifteen questions across Level 6: read piecewise functions off graphs and braces, type the pieces of a graph, then run conditional rules on a mapping diagram.
Open
VII
0 / 6
▶
The Floor Function
Meet the floor function: read it off its staircase, see that it rounds down to the integer at or below, and name the interval of inputs behind each output.
Open
▶
The Ceiling Function
Meet the ceiling function: read it off its beams, see that it rounds up to the integer at or above, and name the interval of inputs behind each output.
Open
▶
Plotting Floor and Ceiling
Build floor and ceiling functions, shifted up or down, so their staircases pass through every given point.
Open
▶
Repeating Functions
Meet functions that repeat: read a sawtooth, describe its pattern, and use the repeat to find many inputs with the same output, on sawtooth and smooth waves alike.
Open
▶
Periodicity
Make repetition exact: a function has period P when f(x) = f(x + P) for every x. Find inputs that share an output, then read the period of waves, sawtooths and a tangent-like curve.
Open
▶
Level Check
Fifteen questions across Level 7: values and periods of repeating functions, floor and ceiling intervals, staircases through points, then a look back at piecewise graphs.
Open
VIII
0 / 4
▶
Domain
Some inputs give a function nothing to work with. Find them, and name the domain: the set of inputs that do give an output.
Open
▶
Range
Which outputs can a function actually produce? Find the input behind an output, spot the outputs that never appear, and name the range.
Open
▶
Domain and Range
Find the domain first, then the range: square roots, a rounding function and a flipped root, with both sets shown on number lines.
Open
▶
Level Check
Fifteen questions across Level 8: inputs with no output, outputs that never come out, domains and ranges as sets and rules, then a look back at points and axes.
Open
IX
0 / 6
▶
Reciprocal Functions
Plot points of f(x) = 1/x, then read its domain and range off the graph: every input except 0, every output except 0.
Open
▶
Squares and Ranges
x² is never negative, so a parabola's outputs stop at its vertex: read the range off the graph and write it as y ≥ or y ≤ a number.
Open
▶
Square Roots and Domains
A negative number has no real square root, so a root function's domain starts where the expression under the root is 0: read it off the graph and write it as x ≥ or x ≤ a number.
Open
▶
Piecewise Ranges
Each piece of a piecewise function makes its own outputs, so the range is everything they make together, gaps and all.
Open
▶
Identifying Domain and Range
Read domain and range straight off a graph: the x-values it covers are the domain, the y-values it reaches are the range.
Open
▶
Level Check
Fifteen questions across Level 9: domains of roots, ranges of parabolas, values a function never reaches and its largest or smallest output, then a look back at number rules.
Open
X
0 / 5
▶
Adding Functions
Add two functions by adding their outputs at every x: plot the sum point by point, then write it as one expression.
Open
▶
Subtracting Functions
Subtract two functions by subtracting their outputs at every x, and see why the order matters: swapping it reflects the graph across the x-axis.
Open
▶
Multiplying Functions
Multiply two functions by multiplying their outputs at every x: a factor can stretch, squeeze or flip the other function, and two lines can multiply into a curve.
Open
▶
Dividing Functions
Divide two functions by dividing their outputs at every x, and find the inputs where the bottom output is 0, so the quotient has no value there.
Open
▶
Level Check
Fifteen questions across Level 10: add, subtract, multiply and divide functions, find where a quotient breaks, then a look back at reading graphs.
Open
0 / 6
▶
Chaining Functions
Apply one function, then another: find each rule from its table, then the single rule that does both, written g(f(x)).
Open
▶
Applying Multiple Functions
Evaluate composites like f(g(3)) from the inside out, rewrite them as f(value), and see that applying the same two functions in the other order can give a different result.
Open
▶
Combining Functions
Find the formula of a composite by replacing x in the outer function with the whole inner function, then simplify.
Open
▶
Graphing Composites
Find a composite's formula and see its graph: f(g(x)) and g(f(x)) usually draw different curves.
Open
▶
Composite Domains
Find the inputs a composite turns away: x is in the domain of f(g(x)) only when g(x) is in the domain of f.
Open
▶
Level Check
Fifteen questions across Level 11: composite formulas, values and domains, typed, then a look back at writing rules as functions.
Open
0 / 7
▶
Vertical Shifting
Add the same number b to every output of f, and the whole graph moves up by b (down when b is negative), shape unchanged.
Open
▶
Horizontal Shifting
Replace x with x + b inside f and the whole graph moves sideways: left by b, because g(x − b) gives what f(x) gave.
Open
▶
Combining Shifts
g(x) = f(x + a) + b moves the graph of f sideways by a (left when a is positive) and up by b, in one combined move.
Open
▶
Vertical Stretching
Multiply every output of f by k and every height scales from the x-axis: k > 1 stretches the graph, 0 < k < 1 compresses it.
Open
▶
Horizontal Stretching
Multiply x by m inside f and the graph squeezes toward the y-axis by m (or stretches away when 0 < m < 1): g(x/m) gives what f(x) gave.
Open
▶
Combining Stretches
g(x) = k · f(jx) stretches the graph of f vertically by k and compresses it horizontally by j, in one combined move.
Open
▶
Level Check
Fifteen questions across Level 12: shifts and stretches typed into formulas, whole graphs moved and stretched, then five translations of points and a shape.
Open
0 / 5
▶
Reflection Across the x-Axis
g(x) = −f(x) reflects the graph of f across the x-axis: every point keeps its x, and its height changes sign.
Open
▶
Reflection Across the y-Axis
g(x) = f(−x) reflects the graph of f across the y-axis: every point keeps its height, and its x changes sign.
Open
▶
Composing Reflections
−f(x) negates the whole expression and reflects across the x-axis; f(−x) replaces every x with −x and reflects across the y-axis.
Open
▶
Reflecting and Stretching
−k·f(x) reflects f across the x-axis and stretches it vertically by k; f(−kx) reflects f across the y-axis and compresses it horizontally by k.
Open
▶
Level Check
Fifteen questions across Level 13: reflections and stretches typed as formulas, whole graphs turned over, then five looks back at composition.
Open
0 / 8
▶
Inverting a Function
The inverse function f⁻¹ undoes f: it sends every output of f back to the input it came from.
Open
▶
Finding the Input
Finding the input that gives an output is the same as applying the inverse function to that output: on a graph, across from the y-value to the line, then down to x.
Open
▶
Finding the Inverse Function
To find f⁻¹(x), rearrange the formula for x, then relabel: x becomes f⁻¹(x) and f(x) becomes x.
Open
▶
Graphing Inverse Functions
The inverse of a linear function is linear too: each point (a, b) of f appears as (b, a) on the graph of f⁻¹.
Open
▶
Reflecting to Invert
Swapping the coordinates of every point, (a, b) → (b, a), reflects the graph of f in the line y = x onto the graph of f⁻¹.
Open
▶
Invertibility
A function whose output comes from more than one x has no inverse; the horizontal-line test finds them: a level line meeting the graph more than once.
Open
▶
Invertible Functions
A function is invertible when every horizontal line meets its graph at most once; a function that fails reflects across y = x into a curve that is not a function.
Open
▶
Level Check
Fifteen questions across Level 14: inverse formulas with and without graphs, counting the inputs for an output, and curves that are not functions.
Open