Definition Calculator

A definition here means the rule that assigns an output to an input: $f(x) = 3x^2 - 5x + 7$ defines $f$, and evaluating it means substituting a number for every occurrence of $x$.

The procedure that never fails:

1. Bracket the substitution. Write $3(-2)^2$, never $3-2^2$. Almost every sign error on this topic comes from an unbracketed negative.
2. Apply the order of operations — brackets, indices, multiplication and division, then addition and subtraction.
3. Keep the negative inside the power where the expression puts it: $(-2)^2 = 4$ but $-2^2 = -4$.

For a function of two variables, substitute both at once and treat the result as arithmetic.

The assumption people forget: a variable stands for the same number everywhere it appears in one expression. If $x$ appears three times, all three become $-2$ — you cannot substitute into some of them and leave the rest.

Indices (exponents) obey a short list of rules, all of which follow from repeated multiplication:

$x^a x^b = x^{a+b}, \qquad \frac{x^a}{x^b} = x^{a-b}, \qquad (x^a)^b = x^{ab}$

$(xy)^a = x^a y^a, \qquad x^{-a} = \frac{1}{x^a}, \qquad x^{0} = 1\ (x \neq 0)$

Fractional indices are roots: $x^{1/n} = \sqrt[n]{x}$, so $x^{m/n} = \left(\sqrt[n]{x}\right)^m$. Take the root first and the arithmetic stays small — $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$.

To solve in terms of $x$, collect every $x$ term on one side, factor $x$ out, then divide:

$ax + bx = c \ \Longrightarrow\ x(a+b) = c \ \Longrightarrow\ x = \frac{c}{a+b}$

When it applies: the division step needs $a + b \neq 0$, and clearing a denominator needs that denominator to be non-zero — state the restriction alongside the answer.

• Losing a bracket on a negative substitution — $(-2)^2 = 4$, while $-2^2 = -4$. They are different expressions.
• Adding indices when multiplying bases — $2^3 \times 3^3$ is not $6^6$. The law $x^ax^b = x^{a+b}$ needs the same base.
• Reading $x^{-2}$ as a negative number — it is a reciprocal, $1/x^2$, and it is positive whenever $x$ is real and non-zero.
• Cancelling across a sum — in $(x+3)/(x+6)$ nothing cancels. Only common factors cancel.
• Forgetting $x^0 = 1$ — including when the base is a whole bracketed expression.
• Dividing by something that could be zero when rearranging — carry the restriction, such as $y \neq 2$, into the final answer.