Meet the nine toolkit functions

Imagine shining light on a graph to cast two shadows. Its shadow on the left-right input axis shows every place an input can lie. That is the domain. Its shadow on the up-down output axis shows every height the graph reaches. That is the range. Read the domain left to right and the range bottom to top. The graph of x$\sqrt{x}$ begins at (0, 0) and continues right and up, so both shadows begin at the included zero. A picture window cuts off a drawing, so use arrows and the formula to decide whether the graph continues beyond it.

• Domain: all horizontal input positions with a point on the graph. Read left to right.
• Range: all vertical heights the graph reaches. Read bottom to top.
• For x$\sqrt{x}$, input 0 works and every positive input works. The domain is [0, ∞).
• For x$\sqrt{x}$, output 0 appears and every positive output y appears at input y²$y^2$. The range is [0, ∞).
• A graph above the horizontal axis has nonnegative outputs. That fact alone does not exclude negative inputs.
• Use braces for separate values, such as {28, 30, 31}. Use an interval for a full stretch, such as [0, ∞).

Imagine a parking lot that charges the same flat fee regardless of how long you stay. Changing the input leaves the output unchanged. A constant is a fixed number, and a constant function gives that fixed number for every input. The formula f(x) = c uses c for the chosen fixed output. It is not a new input. For the example f(x) = 2, input −2, input 0, and input 2 all produce 2. Its graph is a level horizontal line because every point has the same height. Any real input is allowed, but only one output ever appears.

• Formula: f(x) = c, where c is a fixed real number.
• Domain: (−∞, ∞), because the rule accepts every real input.
• Range: {c}, because c is its only output.
• Graph: a horizontal line at height c. The nearby graph uses c = 2.
• Why these restrictions hold: The input x does not appear in the output c, so substituting any real number cannot create a denominator of zero or an invalid root. That gives the whole real-number domain. Since every substitution returns c, no output besides c can appear. Conversely, c really does appear, so the range is exactly the one-value set {c}. Several inputs share c, making the function fail one-to-one.

Imagine a copier that returns exactly the number you hand it. Input 3 comes back as 3, and input −3 comes back as −3. The identity function follows that copying rule: f(x) = x. Nothing is added, multiplied, or erased. Its graph is a diagonal line through the origin, the point (0, 0), because input and output are equal at every point. Every real input is accepted, and every real output can appear by choosing that same number as the input. The output also tells you the original input, so this function is one-to-one.

• Formula: f(x) = x.
• Domain: (−∞, ∞), because copying accepts any real number.
• Range: (−∞, ∞), because input y produces any desired real output y.
• Graph: the diagonal line y = x through (0, 0).
• Why these restrictions hold: Replacing x in f(x) = x merely writes down the selected input, so no real number is excluded. If you want any particular real output y, choose input x = y, and the rule produces it. Thus both domain and range are all real numbers. Two different inputs stay different because the rule copies each unchanged; there is no operation that could merge them into one output.

Imagine measuring how far you stand from the zero mark on a road. Three steps left and three steps right are both a distance of 3. Absolute value is this distance from zero, written |x|. The bars are part of the operation's symbol. The absolute value function f(x) = |x| accepts any real input and returns its nonnegative distance from zero. Its graph has a V shape because either direction away from zero increases the distance. Zero gives zero. An output greater than zero comes from two opposite inputs, so the full function is not one-to-one.

• Formula: f(x) = |x|, distance from zero.
• Domain: (−∞, ∞), because every real number has a distance from zero.
• Range: [0, ∞), because distances are nonnegative and every such distance is reachable.
• Graph: a V with its lowest point at (0, 0). For t > 0, solving |x| = t gives x = −t or x = t.
• Why these restrictions hold: A number can be on either side of zero, but distance has no negative direction. For x ≥ 0, its distance is x; for x < 0, the distance is −x, which turns that negative input into a positive output. Every nonnegative output y can be produced by input y, and a negative output is impossible. These facts give all real inputs as the domain and [0, ∞) as the range.

Picture square floor tiles. If a square has side length 3, it covers 3 × 3 = 9 square units. The toolkit quadratic function squares its input: f(x) = x²$x^2$. Numeric inputs may be negative even though a floor tile's length cannot. Squaring a negative input still gives a positive output because two negative factors multiply to a positive. The graph is a rounded bowl with its bottom at (0, 0). It is called a parabola. You may put in any real number. You can obtain zero or a positive output, but never a negative output.

• Formula: f(x) = x²$x^2$. More generally, a quadratic function is ax²$x^2$ + bx + c with fixed real coefficients and a ≠ 0; its highest power of x is 2.
• Domain: (−∞, ∞), because any real input can be multiplied by itself.
• Range: [0, ∞), because squares are nonnegative and y$\sqrt{y}$ reaches each nonnegative y.
• Graph: a parabola opening upward with lowest point (0, 0).
• Why these restrictions hold: Squaring is multiplication of a real number by itself, which is defined for every real input. Two positive factors give a positive product, two negative factors also give a positive product, and zero squared is zero. Thus no negative output is possible. Every desired nonnegative output y is reached by x = y$\sqrt{y}$, so the range is all of [0, ∞). Opposite nonzero inputs share a square, causing the two sides of the bowl.
• The domain, range, shape and examples here describe the toolkit quadratic x²$x^2$. Other quadratic formulas can have a different lowest or highest height.

Imagine making a cube from blocks. A side of 2 gives 2 × 2 × 2 = 8 blocks. The toolkit cubic function does that three-factor multiplication: f(x) = x³$x^3$. Its numeric inputs can also be negative. Three copies of a negative input give a negative cube, so the sign survives. The graph rises through (0, 0), from negative inputs and outputs to positive ones. Every real input is allowed, and every real output has one cube root. These statements describe x³$x^3$. A different cubic formula can turn around and share outputs.

• Formula: f(x) = x³$x^3$. More generally, a cubic function is ax³$x^3$ + bx²$x^2$ + cx + d with fixed real coefficients and a ≠ 0; its highest power of x is 3.
• Domain: (−∞, ∞), because three-factor multiplication accepts any real input.
• Range: (−∞, ∞), because input ³y$\sqrt[3]{y}$ produces any real output y.
• Graph of the toolkit cubic x³$x^3$: a rising bent curve through (0, 0). This toolkit rule is one-to-one; other cubics may not be.
• Why these restrictions hold: Multiplying three real copies is defined for any real number. A negative times a negative first gives positive, and multiplying by the third negative makes negative again. Thus cubes can be negative, zero, or positive. For any desired real output y, its cube root ³y$\sqrt[3]{y}$ is an input producing y. Cubing keeps increasing as input increases, so two different inputs cannot share the same output, unlike squaring.
• For all-real uniqueness, compare signs and sizes. Larger positive numbers have larger cubes. Among negative numbers, the one farther left has larger positive magnitude, whose cube becomes more negative after restoring the minus. Across zero the signs differ. Thus different inputs of x³$x^3$ give different outputs.

Imagine sharing one whole loaf among x equal shares. For positive x, each share has size 1 divided by x. The reciprocal function uses that division rule numerically: f(x) = 1x$\frac{1}{x}$. It also accepts negative inputs, which give negative outputs. It cannot accept zero because division by zero is undefined, meaning it has no real value. The graph has one branch above and to the right of zero, and another below and to the left. Its output never equals zero: dividing the fixed nonzero numerator 1 by an allowed finite number cannot erase it.

• Formula: f(x) = 1x$\frac{1}{x}$. A reciprocal reverses a nonzero number under multiplication: x × 1x$\frac{1}{x}$ = 1.
• Domain: (−∞, 0) ∪ (0, ∞), because denominator zero is undefined.
• Range: (−∞, 0) ∪ (0, ∞), because the numerator 1 never gives output zero and every nonzero output is reached.
• Graph: two separate pieces on opposite sides of both axes. They approach the axes without touching them.
• The coordinate axes are asymptotes for this rule: as you move far along a branch, its height approaches zero; as x approaches zero, the branch grows without a finite bound. Zero itself is never an input or output.
• Why these restrictions hold: For x ≠ 0, dividing 1 by x produces a real number, so those and only those inputs are allowed. An output of zero would require 1 = 0 × x = 0, which is impossible. Every desired nonzero output y is reached by input x = 1y$\frac{1}{y}$, because taking a reciprocal twice returns the original number. That proves the range also consists of all nonzero real numbers.

Start with the reciprocal picture of sharing one whole, then square the input before dividing. The reciprocal squared function is f(x) = 1x²$\frac{1}{x^2}$. Its denominator is positive for every nonzero input, even a negative one. That makes every output positive. Zero is still forbidden because its square is zero and division by zero has no value. The graph has two separate pieces above the horizontal axis, one on each side of the missing input zero. Opposite inputs share an output because their squares agree. You can get any positive output, but neither zero nor a negative output.

• Formula: f(x) = 1x²$\frac{1}{x^2}$. This also equals (1x$\frac{1}{x}$)²${)}^2$ for x ≠ 0.
• Domain: (−∞, 0) ∪ (0, ∞), because denominator x²$x^2$ is zero exactly when x = 0.
• Range: (0, ∞), because the nonzero square denominator makes every output strictly positive, and each positive output is reachable.
• Graph: two separate pieces above the horizontal axis, with matching heights at opposite inputs.
• Why these restrictions hold: For x ≠ 0, x²$x^2$ is positive, so 1x²$\frac{1}{x^2}$ is positive and defined. At x = 0 the denominator is zero, which excludes that input. To produce any chosen positive output y, set x²$x^2$ = 1y$\frac{1}{y}$ and choose x = 1y$\sqrt{\frac{1}{y}}$, which is nonzero. Its opposite works too. This proves the entire positive range while explaining the graph's matching left and right separate pieces.

Imagine knowing the area of a square tile and asking for its side length. Area 9 gives side 3 because 3²$3^2$ = 9. The square root function f(x) = x$\sqrt{x}$ asks for the nonnegative number whose square is the input. The root symbol selects that nonnegative answer. A negative input cannot be the square of a real number, so it is excluded. The graph begins at the included point (0, 0) and extends rightward and upward. Every nonnegative input works, and every nonnegative output can appear. This graph is one-to-one because each output squares back to one input.

• Formula: f(x) = x$\sqrt{x}$, the nonnegative real square root.
• Domain: [0, ∞). A real square cannot be negative, so negative inputs fail. Every nonnegative input has a nonnegative square root, including input 0 with output 0. Thus all and only nonnegative inputs work.
• Range: [0, ∞), because the symbol chooses the nonnegative root and input y²$y^2$ reaches each y ≥ 0.
• Graph: starts at (0, 0) and curves right and up. It is one-to-one.
• Why these restrictions hold: Every real square is nonnegative, so no real number can square to a negative input. For any input x ≥ 0, the nonnegative square root exists and is the output chosen by the symbol x$\sqrt{x}$. To reach any desired output y ≥ 0, choose input y²$y^2$: y²$\sqrt{y^2}$ = y for nonnegative y. These observations show that both domain and range are [0, ∞), including zero.

Imagine knowing how many blocks fill a cube and asking for its side count. A volume of 8 gives side 2 because 2³$2^3$ = 8. The cube root function reverses cubing: f(x) = ³x$\sqrt[3]{x}$. As an algebraic rule, it accepts negative inputs too. Input −8 gives −2 because three copies of −2 multiply to −8. Its graph bends through zero, extending to negative and positive inputs and outputs. The curve has a different bend from the cubic graph, but their rules undo each other. Every real input has one real cube root, so every real input is allowed.

• Formula: f(x) = ³x$\sqrt[3]{x}$, the unique real cube root.
• Domain: (−∞, ∞), because every real number has a real cube root.
• Range: (−∞, ∞), because input y³$y^3$ produces any desired real output y.
• Graph: a rising cube-root curve through (0, 0), with negative and positive values. It is one-to-one because cubing an output recovers its one input.
• Why these restrictions hold: Cubing produces negative, zero, and positive values without repeating an output. Reversing that operation therefore supplies exactly one real cube root for any real input. Every real output y is reached by choosing input y³$y^3$, since ∛(y³$y^3$) = y. That proves both the all-real domain and all-real range. A negative cube root is allowed because three negative factors have a negative product, unlike two factors in a square.

Think of your exam preparation as packing a small travel bag. Keep the few things you need constantly within reach. Rebuild facts that follow quickly from an operation, and save the full formula table as a reference for study. In a closed-book exam, the reference helps you practice beforehand rather than serving as something you can consult during the exam. Know what a function promises and which direction a question asks you to read. Rebuild domains and ranges by asking whether the operation accepts an input and can reach an output. Use the silhouettes as memory cues, then confirm them with a number.

• Function: each allowed input has one output. One-to-one: each output actually produced has one input. A vertical line is one input. A horizontal line is one output. Check function status first.
• Evaluate f(3): input 3 is given; find its output. Solve f(x) = 3: output 3 is given; find every input. In a table with inputs above outputs, read down to evaluate and up from every matching output to solve.
• Constant: f(x) = c. Domain (−∞, ∞). Range {c}.
• Identity: f(x) = x. Domain (−∞, ∞). Range (−∞, ∞).
• Absolute value: f(x) = |x|. Domain (−∞, ∞). Range [0, ∞).
• Quadratic: f(x) = x²$x^2$. Domain (−∞, ∞). Range [0, ∞).
• Cubic: f(x) = x³$x^3$. Domain (−∞, ∞). Range (−∞, ∞).
• Reciprocal: f(x) = 1x$\frac{1}{x}$. Domain (−∞, 0) ∪ (0, ∞). Range (−∞, 0) ∪ (0, ∞).
• Reciprocal squared: f(x) = 1x²$\frac{1}{x^2}$. Domain (−∞, 0) ∪ (0, ∞). Range (0, ∞).
• Square root: f(x) = x$\sqrt{x}$. Domain [0, ∞). Range [0, ∞).
• Cube root: f(x) = ³x$\sqrt[3]{x}$. Domain (−∞, ∞). Range (−∞, ∞).
• Rebuild restrictions: division excludes zero denominators; real square roots require nonnegative insides. Domain means inputs; range means outputs actually produced. Brackets include finite endpoints; parentheses exclude them. Infinity always uses parentheses.
• Expand: (a + h)²${)}^2$ = a²$a^2$ + 2ah + h²$h^2$. Subtract a whole output by changing every sign. Factor a numerator before canceling a shared nonzero factor.
• Difference quotient: f(x + h) − f(x)h$\frac{f(x+h)-f\left(x\right)}{h}$, or f(a + h) − f(a)h$\frac{f(a+h)-f\left(a\right)}{h}$ when the starting input is named a. Keep h ≠ 0. Compute both outputs, subtract the whole old output, simplify, then divide by the nonzero step h. The result is the average output change for each 1 input step.
• Circle area: A = πr²$r^2$, r > 0. Recover radius with r = Aπ$\sqrt{\frac{A}{\pi }}$. Keep the positive radius and leave π written for exactness.
• Why rebuild: Memorizing every table entry separately hides connections and uses study time poorly. One rule explains several facts: a denominator cannot be zero, a square is nonnegative, and a cube keeps the sign. Rebuilding these restrictions from their operations guards against mixing up domain and range. A compact reference collects the formulas for practice, while the definitions and question directions need to be available immediately for every problem.

Keep these decisions available from memory. They tell you what kind of question you are answering before you begin calculating. Each memory cue appears with a worked example and a counterexample in its own lesson.

• Know cold: a function gives one output per accepted input. Memory cue: one arrow OUT of each input. See the function lesson.
• Know cold: in f(3), 3 is the input; in f(x) = 3, 3 is the output. Memory cue: inside names the input, equals names the output. See function notation.
• Know cold: a vertical line is one input; a horizontal line is one output. Use the vertical test for function status first, then the horizontal test for one-to-one.
• Know cold: wrap the whole substituted input in parentheses. Memory cue: the input's seat belt.
• Know cold: the difference quotient is new output minus old output, divided by the step h. Keep h ≠ 0. See the difference-quotient lesson.
• Know cold: domain is input and range is output. Memory cue: D before R, I before O, x before y.
• Know cold: the nine names and silhouettes. Memory cue: flat fee line, copy line, distance V, square bowl, cube bend, reciprocal two pieces, reciprocal-square two upper pieces, and roots undo powers.

Practice the operations that produce an answer. You do not need to memorize each finished expansion, factorization or domain. Rebuild them from multiplication, division and the meaning of a root. The refreshers teach these moves before the later lessons ask you to use them.

• Understand, then rebuild: expand a squared sum by multiplying two copies. (x + h)²${)}^2$ = x²$x^2$ + 2xh + h²$h^2$.
• Understand, then rebuild: find factor pairs by product and sum, then distribute to check.
• Understand, then rebuild: isolate the requested output with equal operations on both sides.
• Understand, then rebuild: compute a difference quotient by substituting x + h, subtracting the whole old output, simplifying and dividing by nonzero h.
• Understand, then rebuild: sketch toolkit graphs from helpful points and their silhouettes, then use operation restrictions to confirm their full domains and ranges.

The final lesson gathers a short printable reference with the toolkit formulas and the main procedures. Use it during study and when reference material is allowed. For a closed-book exam, close it and practice rebuilding each restriction from the rule's meaning.

• Put on the cheat sheet, or look it up during study: the nine-formula table above, with domain, range, one-to-one status and each graph silhouette.
• Put on the cheat sheet, or look it up during study: f(x + h) − f(x)h$\frac{f(x+h)-f\left(x\right)}{h}$, h ≠ 0, and its step-by-step procedure. Know the meaning from memory for a closed-book exam.
• Put on the cheat sheet, or look it up during study: interval brackets include finite endpoints; parentheses exclude them; infinity always takes parentheses.
• Put on the cheat sheet, or look it up during study: circle area A = πr²$r^2$ for r > 0 and the reverse radius rule r = Aπ$\sqrt{\frac{A}{\pi }}$.
• A closed-book exam does not allow this reference at the desk. Practice with it, then cover it and rebuild the facts from the operations.