Composition of functions

A day determines a temperature, and that temperature determines a heating cost, so one function's answer can become another function's input. This section first compares arithmetic on two function outputs with composition, which connects functions in order. You will follow that connection through tables, graphs, and formulas, then check both stages to find the allowed starting inputs. Finally, you will split a complicated formula into the jobs that build it. The practice covers the kinds of questions used in section 1.4: arithmetic combinations, compositions in either order, tables, graphs, domains, and splitting a formula.

Lessons

Vocabulary

Function: A rule that gives exactly one output for each input it accepts. Like: A vending machine with one result for each accepted button press.
Input: The starting value you give a function. It goes inside the parentheses in function notation. Like: The ingredient you put into a machine.
Output: The answer a function produces from an accepted input. Like: The item that comes out of a vending machine.
Evaluate: Find the output of a function or the value of an expression for a given input. Like: Run the machine for one chosen starting value.
Algebraic operations: The familiar actions of addition, subtraction, multiplication, and division, applied to numbers or expressions. Like: Four ways to combine amounts of money.
Sum of functions: A new function that adds two outputs produced from the same input. Both original functions must accept that input. Like: Add two people's incomes for the same year.
Difference of functions: A new function that subtracts one output from another at the same input. Subtract the entire second expression. Like: Subtract expenses from income for the same month.
Product of functions: A new function that multiplies two outputs produced from the same input. Both original functions must accept that input. Like: Multiply one day's item count by that day's price per item.
Quotient of functions: A new function that divides two outputs at the same input. Both outputs must exist, and the bottom output must not be zero. Like: Divide a day's total cost by that day's number of items.
Ratio: A comparison made by division. The first amount is divided by the second, so the second must not be zero. Like: Compare three cups of water to two cups of juice.
Domain: All inputs for which a function is defined. Like: The list of coins a machine accepts.
Range: All outputs a function actually produces from its allowed inputs. Like: The set of items a vending machine can dispense.
Undefined: Having no permitted output under the rule being used, as with division by zero or a negative real square-root radicand. Like: A machine cannot complete the requested job.
Composition of functions: Using the output of one function as the input of another function. Like: One machine hands its finished item directly to the next machine.
Composite function: The new function produced by composition. Its single rule carries an input through both original functions in order. Like: A complete route through two connected machines.
Composition operator: The open-circle symbol ∘ that joins two function names to indicate composition. Like: A connector showing which two machines belong in the route.
Binary operation: An operation that takes two starting objects and produces one result. Here binary means two objects, rather than the computer digits 0 and 1. Like: A recipe that takes two ingredients and makes one mixture.
Inner function: The function evaluated first in a composition. Its output becomes the next function's input. Like: The first machine on the production line.
Outer function: The function evaluated second in a two-function composition. It receives the inner function's output. Like: The second machine receives the first machine's finished item.
Commutative: Describing an operation whose result stays the same when the order is reversed. Function composition usually does not have this property. Like: Two chores whose order does not change the finished result.
Unit: The named measure attached to a quantity, such as hours, miles, gallons, or dollars. Like: A label on a measuring cup tells you cups or milliliters.
Substitution: Replacing a variable with a number or expression everywhere that variable appears. Parentheses hold a multi-term replacement together. Like: Put the same chosen ingredient into every slot labeled for it.
Domain of a composite: Inputs the inner function accepts and whose inner outputs the outer function accepts. Like: Admission requires passing the first gate and then the second gate.
Radicand: The number or expression inside a root symbol. A real square root requires its radicand to be nonnegative. Like: The contents underneath the root's roof.
Square root: The nonnegative number whose square is the given nonnegative number. Here the √ symbol names that one nonnegative value. Like: A square tile's side length when you know its area.
Reciprocal: One divided by a nonzero number. A fraction's reciprocal swaps its top and bottom, and their product is 1. Like: Turn a nonzero fraction upside down.
Decomposition: Writing a function as a composition of simpler functions by naming an inside job and an outside job. Like: Split one cooking task into preparation and cooking.
Component functions: The individual functions that make up a composite function. Like: The separate stations that make up one production line.
Interval notation: A way to write stretches of the number line. Brackets include finite endpoints; parentheses exclude endpoints and always accompany infinity. Like: A map marking a road segment and whether its end stops belong.
Intersection: The values shared by all the sets or conditions being combined. Like: Guests whose names appear on both invitation lists.
Absolute value: A number's distance from zero, written with vertical bars. Distance is never negative. Like: How many steps you are from home, regardless of direction.
Ordered pair: Two coordinates written (x, y). The first gives horizontal position and the second gives vertical position. Like: An address that says across first, then up or down.
x-axis: The horizontal number line of a coordinate graph. A function's inputs are read here. Like: The east-west line on a map.
y-axis: The vertical number line of a coordinate graph. A function's outputs are read here. Like: The north-south line on a map.
Function composition: Another name for composition of functions: one function's output becomes another function's input. Like: Connect the exit of one machine to the entrance of another.
Formula: A written rule using numbers, symbols, and operations to describe how a quantity is calculated. Like: A recipe written with letters for the amounts.
Variable: A symbol that stands for a number or quantity. Its value may change or may be the unknown you need to find. Like: An empty labeled box waiting for a chosen value.
Table: An organized display of matching values. In a function table, each input and its output share a column or row. Like: A list of addresses paired with their destinations.
Graph: A picture of coordinate pairs. A function graph shows each input horizontally and its output vertically. Like: A map of the rule's input-output addresses.
Denominator: The bottom of a fraction. It names the divisor and cannot be zero. Like: The piece-size label on a share of a pizza.
Real number: A number on the ordinary number line, including negatives, zero, fractions, and numbers such as $square root of 2$ . Like: Any possible signed location along one straight measuring line.
Order of operations: The agreed calculation order: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right. Like: A recipe's instructions tell you which job comes first.
Nonnegative: Greater than or equal to zero. Zero is included, while positive means greater than zero and excludes zero. Like: A bank balance that is not in debt, including an empty account.
Union: The values that belong to at least one of the sets being combined. The symbol is ∪. Like: Guests invited by either of two hosts.
Difference of squares: One squared expression minus another. It factors into the first expression minus the second, times the first plus the second. Like: Two opposite middle pieces cancel when the product is rebuilt.
Factor: One of the quantities multiplied together in a product. Like: One building piece that joins another by multiplication.
Factoring: Rewriting an expression as a product of factors without changing its value. Like: Unpack one amount to see the pieces multiplied inside it.
Cancellation: Dividing a fraction's top and bottom by the same nonzero factor. It preserves the value on the original domain. Like: Repackaging equal shares into larger pieces without changing the amount.
Binomial: An algebraic sum or difference with two terms, such as x + 2 or 3x − 1. Like: A package with two separate parts inside.
Fraction: One number divided by another nonzero number. The top is divided by the bottom; equal pieces give a picture of its size. Like: Three slices from a pizza cut into four equal slices.
Exponent: A raised number indicating a power. A positive whole-number exponent counts how many copies of the input are multiplied together. Like: A count telling you how many identical factors to use.
Intermediate answer: The in-between output of the first function, which becomes the next function’s input. Like: A connecting station between two trips.
Nested: Placed inside another expression, like one pair of parentheses inside another. Like: Boxes fitted inside boxes.
Linear: A formula built from a constant times the input plus a constant. Its graph is a straight line. Like: A steady-rate trip on a straight road.
Quadratic: A formula built from a nonzero number times $x^{2}$ , a number times x, and a constant, added together. Its graph is a U shape opening upward or downward. Like: A bowl opening upward or downward.
Constant function: A function that gives the same output for every input it accepts. Like: A machine that always dispenses the same item.
Hole: One missing graph point, shown with an open dot. Its input does not have that plotted value. Like: A gap at one place in a road.
Descriptive variable: A letter chosen to remind you what it measures, such as d for day, T for temperature, or C for cost. Like: A name tag on a jar.
Solve: Find every input that makes a stated equality or inequality true. This differs from evaluating at a given input. Like: Find which button produces the requested item.
Endpoint: A boundary number at an end of an interval. A bracket includes it; a parenthesis leaves it out. Like: A road’s end stop.
Set-builder notation: A set written by describing the condition its members must satisfy. The bar reads such that. Like: An admission list described by its entry requirement.
Like terms: Terms with the same variable part and powers. Their number factors can be added or subtracted. Like: Combine matching kinds of items in a grocery count.
Distribute: Multiply each term inside parentheses by the outside factor. Like: Each of three bags contains every listed item.
Equation: A statement that two expressions have equal values. Like: A balanced scale with equal amounts on both sides.
Inequality: A comparison saying less than, greater than, or either relation with equality allowed. It can describe many permitted inputs rather than one number. Like: A minimum age or maximum capacity.
Numerator: The top of a fraction. It is the amount divided by the denominator. Like: The count of equal pieces you have.
Power: The result of repeatedly multiplying a number or expression by itself. A positive whole-number exponent counts the copies. Like: Make the stated number of identical multiplied copies.

Quick checks

Let f(x) = + 2 and g(x) = 4x. Find f(g(1)) and g(f(0)). You want the output of each two-step route.

f(g(1)) = f(4) = 18, because g acts first.
g(f(0)) = g(2) = 8, because f acts first.

Use the two pictured complete tables. Find p(q(2)); then find every start x for which p(q(x)) = 5. The first gives an input and asks for an output; the second gives an output and asks for inputs.

p(q(2)) = p(3) = 4.
p(q(x)) = 5 requires q(x) = 1, so x = 1 or x = 4, because only p(1) is 5 and the q output 1 appears in those two columns.

p(t) counts pages printed in t minutes, and w(n) gives grams of ink used for n pages. What does w(p(8)) measure, and why does p(w(8)) lack the stated meaning?

w(p(8)) measures grams of ink used in 8 minutes of printing, because pages are the shared handoff.
p(w(8)) passes grams to a rule expecting minutes, so its units do not match.

Let f(u) = and g(x) = . Find the domain of f(g(x)). You want starts where the root exists and its answer does not make the outer bottom zero.

[−11, 5) ∪ (5, ∞), because x ≥ −11 makes the root real and $square root of x + 11$ = 4 gives x + 11 = 16, hence excluded start 5. At 5 the root is 4 and the outer bottom is 0
at −11 the outer bottom is −4 and works.

Let f(x) = 5x − 2. Find f(f(x)). You want to feed the first answer into the same rule again.

f(f(x)) = 5(5x − 2) − 2 = 25x − 12, because the second application acts on the entire first answer, rather than squaring that answer.

Let g(x) = x − 7 and f(u) = . Find the domain of f(g(x)). You need the inner output to be allowed under the root.

[7, ∞), because x − 7 ≥ 0 gives x ≥ 7 and is defined at the endpoint.

Now use g(x) = x − 7 and f(u) = . Is input 7 still allowed? Find the domain.

Input 7 is excluded.
Domain: (7, ∞), because the denominator $square root of x − 7$ must be real and nonzero, requiring x − 7 > 0.

The pictures show g(x) = 6 − x and f(x) = with no answer labels. Read f(g(2)) from the grid. You want the height on g at input 2, then the height on f at that new input.

g(2) = 4, then f(4) = 16, because the first height becomes the second horizontal input.

Before you start

Explain it like I am five: pass one answer to the next machine
Picture two machines connected by a short belt. A machine is a rule that takes a starting number, called its input, and gives one answer, called its output. You send 2 into a machine that adds 1. It sends out 3.
The belt carries that 3 into a second machine that doubles its input. Doubling means adding a number to itself, so 3 becomes 3 + 3 = 6. The second machine receives the first answer.
This connection is called composition. It is useful when one answer supplies information the next job needs. To find a heating bill for a day, you first find that day's temperature. You then hand the temperature to the heating-cost rule. The starting day and the final dollar amount are linked through the temperature.
Function notation and replacing an input
Picture a vending machine with one item for each accepted button. A function works like that: you choose an input, the rule acts, and you receive one output. A number machine can say to double the input and add 1. With input 3, doubling means 3 + 3 = 6, and the output is 7. The written recipe f(x) = 2x + 1 is a formula. f names the recipe; x names its input slot. Read f(3) aloud as f of 3. Evaluate means find the output. Substitution means replace the input letter wherever it occurs.
Signed numbers and order of operations
Picture your bank balance. Adding 5 dollars raises it by 5. Subtracting 5 dollars lowers it by 5. Removing a debt raises it, which is why subtracting a negative adds: 2 − (−3) = 5. Multiplying by a negative reverses direction; two reversals restore the original direction, so a negative times a negative is positive. Order of operations is the agreed reading order for a formula. The memory cue PEMDAS means parentheses, exponents, multiplication and division, addition and subtraction. The paired operations share a level and go left to right. A positive whole number used as an exponent counts the copies you multiply: $3 squared$ means 3 × 3.
Distribute a factor and subtract a whole expression
Imagine buying two identical bags. Each bag contains 3 apples and 1 orange. Two bags contain 2 × 3 apples and 2 × 1 orange. A factor is a number or expression multiplied by another amount. Distributing means multiplying every part inside a group by the factor outside it. A minus sign before a group means subtract the entire group. You can think of it as multiplying the group by −1. That changes the sign of every piece inside, including a piece that was already negative. Keep the group in parentheses until each part has received the outside factor. A term is one piece joined to other pieces by addition or subtraction, such as 3x or −2 in 3x − 2.
Multiply two groups and expand a square
When x is positive, a rectangle with side lengths x + 2 and x + 3 can be cut into four smaller rectangles. Their areas are x × x, x × 3, 2 × x, and 2 × 3. Adding those four areas gives the whole area. Multiplying two groups follows the same pattern: every term in the first group multiplies every term in the second. A group with two terms is called a binomial. Squaring a binomial means multiplying the entire group by another copy of itself. It creates two matching middle pieces, so those pieces must be included.
Replace an input with a whole expression
Think of one restaurant order held together in a bag. An input can be one number or a whole expression. Keep the whole replacement in parentheses, as you would keep the entire order together before processing it. Every copy of the old input letter gets the same bag. A formula with two input slots needs that replacement twice. This is substitution with an expression, and the previous refreshers have now taught the distribution and powers needed to carry it out.
Fractions, reciprocals, and division
A fraction is a division: $3 over 4$ means 3 ÷ 4, or three equal quarter-pieces of one whole. The denominator tells how many equal pieces make one whole; the top counts those pieces. In $3 over 4$ , each piece is one quarter, and you have three of them. To add pieces, first make their sizes match. To multiply fractions, multiply the tops and multiply the bottoms. A reciprocal is the upside-down version of a nonzero number, such as $2 over 3$ and $3 over 2$ . Dividing by a fraction asks how many of those pieces fit into your amount. It gives the same answer as multiplying by the fraction's reciprocal. A zero bottom makes division undefined, meaning it has no permitted numerical answer: no number times 0 gives 1, and every number times 0 gives 0, so $0 over 0$ has no unique answer. The sign ≠ means not equal to.
Factor a difference of squares and cancel safely
Factoring reverses multiplication. You turn one expression into a product of smaller pieces. Each multiplied piece is a factor. A difference of squares means one squared quantity minus another, such as $x^{2}$ − $1 squared$ . It becomes (x − 1)(x + 1). Think of unwrapping a package to see the matching pieces hidden inside. A fraction can lose a matching factor from its top and bottom only when that factor is not zero. Record the original forbidden input before cancellation. A shorter formula cannot repair a division that was impossible at the start. The accepted-input list is called the domain. Undefined means the calculation has no permitted numerical answer, as with a zero bottom.
Solve a linear equation or a fraction equation
An equation says the two sides have the same value. Picture a balanced scale. Adding the same amount to both pans keeps it balanced; dividing both amounts by the same nonzero number does too. Solving means finding the input that makes the equality true. A linear equation has the unknown to the first power, such as 3x − 2 = 4. A fraction equation may become linear after you multiply by its denominator. Record where that denominator is zero first, because an illegal input can never become a solution.
Real square roots, nonnegative outputs, and absolute value
Think of a square tile with area 9. Its side length is 3, because 3 × 3 = 9. A square root names that nonnegative side length. Nonnegative means zero or greater, so $square root of 0$ = 0 belongs with the positive roots. The number inside a root is the radicand. A real number is a number on the ordinary number line. Every real square is nonnegative, so a negative radicand has no real square root. Absolute value means distance from zero, so |−3| = 3 and |3| = 3. The symbol > means greater than, < means less than, ≥ means greater than or equal to, and ≤ means less than or equal to. Equality is included only in the last two comparisons. An even root undoes a power of 2, 4, or another positive even whole number. Even means divisible by 2 with no remainder. A square root undoes a power of 2, and a fourth root undoes a power of 4. These roots require a nonnegative radicand for real values. The sign ≈ means approximately equal to. A rounded decimal is a nearby value; the radical keeps the exact value.
Solve inequalities, including squares and roots
An inequality compares sizes instead of saying they are equal. On a number line, greater numbers stand farther right. Adding the same number shifts both positions equally and keeps their order. Multiplying both by a positive number changes their positions without reversing order. Multiplying or dividing by a negative reflects them across zero, so the comparison sign reverses. A square condition asks about distance from zero. A root condition needs a real root first. You may square both sides to remove a root only after checking that both sides are nonnegative.
Solve a square-root equation
Picture a square tile whose side must be 4. Its area must be 4 × 4 = 16. A square-root equation asks which inside amount gives a chosen root output. Squaring both sides undoes a square root when its requested output is zero or more. A negative requested output cannot work, because the root symbol always chooses a nonnegative value. After squaring, check the answer in the original root equation, the same way you would check the original tile size after a measurement.
Domain, intervals, union, and intersection
Think of a machine's accepted-input list. Its domain is every input for which it can produce an answer. Its range is the outputs it actually produces. Interval notation packages a stretch of the number line: a bracket includes an endpoint and a parenthesis leaves it out. Infinity describes an unending direction, not a number you can include. Union means keep points in either allowed piece. Intersection means keep only points shared by all conditions. Arithmetic combinations need both functions to accept the same starting input, so their restrictions meet by intersection. A finite endpoint is an ordinary number, rather than infinity.
Read a table downward or upward
A table is a list of input-output addresses arranged in matching columns. The top cell tells you what enters; the bottom cell tells you what comes out. Read down a column to evaluate a function at a given input. If you are given an output and asked which inputs make it, scan the bottom row and read upward from every match. For this refresher, the table lists every input this function accepts. It accepts only those shown inputs. That statement matters: a partial table would leave unlisted inputs unknown rather than prove them undefined.
Read graph coordinates as input and output
A graph is a picture of input-output addresses. Its horizontal line is the x-axis, where you read inputs. Its vertical line is the y-axis, where you read outputs. An ordered pair (2, 5) means 2 across and 5 up, so it shows input 2 giving output 5. Think of finding an address on a map: travel across first, then up or down. To evaluate from a graph, start at the input on the x-axis, move vertically to the curve, then read that point's height on the y-axis.
Read a percentage discount
Picture a price split into 100 equal parts. Percent means out of 100. A 25% discount removes 25 of those parts and leaves 75 of them. You can find the price that remains by multiplying by $75 over 100$ , which is 0.75. A fixed coupon removes a stated dollar amount instead. These are different kinds of instructions, so a store applying both instructions may get different answers when their order changes.