Evaluating means replacing every occurrence of a letter with the same number and then doing arithmetic under the rules of “Order of Operations”. One discipline removes almost every error: write the substituted value inside brackets, every time, whether it looks necessary or not.
Core ideas
Substitution looks like the easiest thing in the course, and it produces wrong answers at a rate that says otherwise. The cause is always the same: the value went in without brackets.
Everything else about the expression stays exactly as it was — which operations, in which order, with which brackets. Putting the value in brackets is how you guarantee that, because a negative value squeezed in bare can change what the next operator applies to.
In x² − x at x = −3, both xs are −3. A letter names one number throughout a single evaluation; if two quantities can differ, they need two letters.
1/(x − 2) has a value at every x except 2, where the substitution produces a division by zero. "Undefined" is a legitimate and complete answer, and Lines, Functions and Graphs gives the set of numbers that do work a name.
The material
One discipline, and the three places it is what saves you.
To evaluate an expression at given values is to replace every occurrence of each variable by its value and then carry out the arithmetic, using the order of operations from “Order of Operations”.
The result is a single number, or else a statement that the expression has no value there.
The discipline: write the value inside brackets. Not as a precaution to be dropped once you are confident — as the method. The brackets say that the whole value, sign included, is what the surrounding operation applies to, and without them the sign can escape.
The second line is not a slip of the pen; it is a correct evaluation of a different string. In −3² the exponent binds more tightly than the minus sign, so it means −(3²). “Order of Operations” settled that, and substitution is where the ruling has consequences.
Compare two expressions that differ only by brackets. At x = −3:
Both evaluations are right. They are answers to different questions, and neither is a misreading of the other. In −x² the minus is applied after the squaring, so a negative x gives a negative result; in (−x)² the negation happens first and the square is positive. Knowing which expression you were handed is the whole of it.
3x means 3 · x, so at x = 4 it is 3(4) = 12 and never 34. This is the clearest reason to bracket the substituted value: without brackets the multiplication sign that was invisible when there was a letter there becomes invisible when there is a digit there, and digits written next to each other mean something else entirely. A coefficient is also not part of the base: 2x² is 2·(x²), because the exponent attaches only to the x immediately under it.
With more than one letter, substitute all of them before doing any arithmetic, then evaluate the resulting numerical expression in the usual order:
The third line is not a failure of the method. Division by zero has no value, so the expression has no value at x = 2, and the correct answer to "evaluate it there" is that you cannot. Every other real number works. Lines, Functions and Graphs calls the set of inputs that work the domain and makes finding it a routine step.
2x² − 3x + 1 at x = −1/2. In brackets: 2(−1/2)² − 3(−1/2) + 1. The square is 1/4, so the first term is 2 · 1/4 = 1/2. The second is −3 · (−1/2) = 3/2. Then 1/2 + 3/2 = 2, and 2 + 1 = 3. The value is 3 — exact, with no decimal anywhere in the working.
Interactive
Brackets round the value, every time
Method
Four steps, and the first one is the only one that is a habit rather than arithmetic.
(−3), not −3. Do this for every occurrence of every letter, and change nothing else about the expression while you do it.
3x becomes 3(4), which is a product. If you find yourself looking at two digits written next to each other, a multiplication sign has gone missing.
Brackets, then exponents, then multiplication and division left to right, then addition and subtraction left to right. “Order of Operations” gave the rule; substitution does not amend it.
A division by zero, or an even root of a negative number, means the expression is undefined at that input. Write that down rather than producing a number.
The first is a fraction; the second is the sign trap.
In part 1 the sign work is all in the second multiplication: −3 times −1/2 is +3/2, and dropping that sign gives 0 instead of 3. Everything stays an exact fraction, so nothing is lost to rounding.
Part 2 shows two expressions and two answers, not one expression and a choice. Which one you are evaluating is decided by the brackets printed in the question, and the only way to keep them straight is to carry them through the substitution.
Fade the substitution for 5 − 3x at x = −2: copy it as 5 − 3(−2) and finish from there. The value is 11. An answer of −1 dropped the substituted sign; 16 treated 3x as the two-digit number 3 followed by 2 before subtracting.
Evaluate 4x² − 3xy + y² at x = −1/2, y = −3, writing every substitution in brackets and keeping exact fractions. Then explain how −x², (−x)² and −2x² differ before evaluating any of them. Lines, Functions and Graphs repeats this act under the name f(x).
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