Evaluate a formula

Explain it like I am five

Think of a recipe with a blank for an ingredient amount. You choose a number and put that same number in every copy of the blank. A function formula works that way: evaluating means the input is known and you want the output. Parentheses are a seat belt around the whole input. They keep a negative sign or a sum attached while you square or multiply. If the input is a letter, you make a recipe for any value of that letter. If the input is a sum, put the entire sum in every blank, then multiply out and collect matching terms.

Squaring the negative input gives 9, and three times that input gives −9.

Reminder

Signed arithmetic. A negative squared is positive when enclosed: (−2 $) squared$ = 4, while 3(−2) = −6.
Distributive property. Multiply every term inside: 3(a + h) = 3a + 3h.
Squaring a sum. (a + h $) squared$ = $a^{2}$ + 2ah + $h^{2}$ because it contains four products, not two.
Subtracting parentheses. −( $a^{2}$ + 3a − 4) = − $a^{2}$ − 3a + 4; each sign changes.
Nonzero factor cancellation. $h(2 a + h) over h$ = 2a + h only for h ≠ 0.

Why it works. A formula tells how an output is built from its input. Replacing every occurrence of the input variable by the same value applies that definition consistently. Replacing only one occurrence changes the recipe. Parentheses preserve the input as a whole: (−3 squares −3, whereas − negates the square of 3. For sums, distribution rewrites the result without changing its value, because each term multiplies every relevant term in the other group.

RuleTo evaluate a formula, replace every input variable with the whole stated input in parentheses, then calculate or expand. The value or expression you obtain is the output.

The same idea, five ways

Say it

f of negative three

Write it

The input is −3. Find the output of f at that input.

In math

f(−3)
f(−3) = (−3 $) squared$ + 3(−3) − 4 = −4

Fill every ingredient blank with the same amount.

See it

Squaring the negative input gives 9, and three times that input gives −9.

The same idea, other ways

Follow a recipe

For f(x) = $x^{2}$ + 3x − 4, the recipe is square the input, add three times the input, then subtract 4. Input 2 gives 4 + 6 − 4 = 6. The name f stays on the machine; 2 is the input and 6 is the output.

Every use of the input in this recipe receives the same number, 2.

Fill every blank

Read $x^{2}$ + 3x − 4 as (blank $) squared$ + 3(blank) − 4. If the input is a + h, write (a + h) in both blanks. This makes f(a + h) = (a + h $) squared$ + 3(a + h) − 4 before any expansion. The parentheses act like a bag keeping all of the input together.

(input + 3(input) − 4

input = a + h

(a + h + 3(a + h) − 4

The entire expression a + h replaces each x.

See the four pieces of a square

A square with side a + h splits into four rectangles: $a^{2}$ , ah, ha and $h^{2}$ . The two middle pieces add to 2ah. That is why (a + h $) squared$ = $a^{2}$ + 2ah + $h^{2}$ . The picture uses positive lengths, while multiplication gives the same identity for any real a and h.

There are two middle rectangles, so the expansion needs 2ah.

.1Numeric substitution

Think of a recipe with two places asking for the same ingredient amount. A function such as f(x) = $x^{2}$ + 3x − 4 uses the input twice, so you must place your number in both spots. A negative input belongs in parentheses. The square uses the whole negative number, while the separate multiplication by 3 keeps its negative sign. Follow the order of operations after replacing the input.

Evaluate means input is known and output is wanted.
f(−3) uses (−3 $) squared$ , which is 9. Writing − $3 squared$ without parentheses means −( $3 squared$ ), which is −9.

Squaring the negative input gives 9, and three times that input gives −9.

Reminder

Order of operations. Powers and products come before sums: $2 squared$ + 3 × 2 = 4 + 6 = 10.

The same idea, five ways

Say it

f of negative three equals negative four.

Write it

Replace both input positions by the whole negative input, then compute the output.

In math

f(x) = $x^{2}$ + 3x − 4
f(−3) = (−3 $) squared$ + 3(−3) − 4 = −4
(−3, −4)

Put the same ingredient amount in every matching blank in a recipe.

See it

Squaring the negative input gives 9, and three times that input gives −9.

Worked examplePositive and negative inputs

For f(x) = $x^{2}$ + 3x − 4, evaluate f(2) and f(−3). Both questions give an input and ask for its output.

Squaring the negative input gives 9, and three times that input gives −9.

What it asks. Use each stated input to find its output from the named formula.

Plan. Replace every input-variable slot with the whole input in parentheses. Work powers before multiplication, and multiplication before adding or subtracting. For a square root, compute its inside first and choose the nonnegative root. Check by following the original recipe again.

Every input slot receives the complete input, including its sign.
Square a negative input in parentheses: (−2 $) squared$ = 4. A constant recipe gives the same output for every input.
Keep separate requested outputs on separate answer lines.

f(2) = (2 $) squared$ + 3(2) − 4.The input 2 replaces every x in the formula.
f(2) = 4 + 6 − 4 = 10 − 4 = 6.Do the square and multiplication before adding and subtracting.
f(−3) = (−3 $) squared$ + 3(−3) − 4.Parentheses keep the negative sign attached to the whole input.
f(−3) = 9 − 9 − 4 = 0 − 4 = −4.A negative times a negative is positive, but 3 times a negative is negative.

Answer

f(2) = 6.
f(−3) = −4.

Check At input 2, the pieces 4, 6, and −4 total 6. At input −3, the first two pieces are 9 and −9, so they cancel and leave −4.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: f(−3) = − − 9 − 4 = −22.

− means negate the square of 3. The formula asks to square the entire input −3 instead.

✓ Instead: f(−3) = (−3 + 3(−3) − 4 = 9 − 9 − 4 = −4.

Tips and tricks

Draw parentheses around every replacement before doing arithmetic.
Two blanks, one number: Write f(2) = (2 $) squared$ + 3(2) − 4. Both blanks receive 2, as both copies of your name on a form receive the same name.
Signed multiplication: For input −3, the square is (−3)(−3) = 9, while 3(−3) = −9. Compute these pieces separately so the two signs cannot get mixed together.

.2Symbolic substitution

A letter can stand for an amount you have not chosen yet, like an empty measuring cup marked a. Replacing x with a gives a recipe for any value of a. Replacing x with a + h puts the entire sum into each input spot. You can leave a correct answer unexpanded, or use the distributive property, multiplying each piece, to rewrite it. Expansion changes the appearance of the output expression, not the value it will have when numbers are chosen.

An algebraic expression can be an input: f(a + h) means use the whole sum a + h.
(a + h $) squared$ = (a + h)(a + h) = $a^{2}$ + ah + ha + $h^{2}$ = $a^{2}$ + 2ah + $h^{2}$ .
3(a + h) = 3a + 3h because 3 multiplies each term in the sum.

The square of a + 1 contains , two copies of a, and 1.

Reminder

Subtracting parentheses. A subtraction applies to every term: 7 − (a + 1) = 7 − a − 1.

The same idea, five ways

Say it

q of the whole sum a plus one equals a squared plus a.

Write it

The complete expression a + 1 replaces each input variable before the result is expanded.

In math

q(x) = $x^{2}$ − x
q(a + 1) = (a + 1 $) squared$ − (a + 1)
q(a + 1) = $a^{2}$ + a

Keep a sealed ingredient bag together when filling each recipe blank.

See it

The square of a + 1 contains , two copies of a, and 1.

Worked exampleAn expression input with a subtraction

For q(x) = $x^{2}$ − x, find q(a + 1). The whole input is a + 1; find the output expression.

q(a + 1) = (a + 1 − (a + 1).

(a + 1 = (a + 1)(a + 1) = + a + a + 1 = + 2a + 1.

q(a + 1) = + 2a + 1 − a − 1.

q(a + 1) = + a.

Read these successive steps along with their reasons.

What it asks. Use the entire expression as input and find the simplified output expression.

Plan. Replace every input-variable slot with the whole input in parentheses. Expand a squared sum with all four products and distribute multiplication or subtraction through the entire group. Combine matching terms and check by assigning numbers to the input letters.

Every input slot receives the complete input, including its sign.
A square of a sum includes two middle products. A negative sign outside parentheses changes every enclosed sign.
Keep separate requested outputs on separate answer lines.

q(a + 1) = (a + 1 $) squared$ − (a + 1).Both copies of x receive the entire input a + 1.
(a + 1 $) squared$ = (a + 1)(a + 1) = $a^{2}$ + a + a + 1 = $a^{2}$ + 2a + 1.Multiply every term in the first parentheses by every term in the second.
q(a + 1) = $a^{2}$ + 2a + 1 − a − 1.Subtracting the whole input changes the signs of both a and 1.
q(a + 1) = $a^{2}$ + a.2a − a = a and 1 − 1 = 0.

Answer

q(a + 1) = + a

Check Choose a = 2. The input is 3, so the original formula gives q(3) = − 3 = 6. The new expression gives + 2 = 6 too.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: (a + h = + .

This leaves out the products ah and ha. With a = 1 and h = 2, the left side is 9 while the proposed right side is 5.

✓ Instead: (a + h = + 2ah + , giving 1 + 4 + 4 = 9 in that check.

Tips and tricks

Write the squared sum as two identical parentheses before expanding; the middle term becomes visible.
A sealed input bag: Place a + h inside parentheses before following the recipe. You are squaring the bag's whole contents, not squaring a and then adding h.
Multiply every pair: In (a + h)(a + h), the first a multiplies a and h, then the first h multiplies a and h. The four products are $a^{2}$ , ah, ha and $h^{2}$ . None can be skipped.

Strategy: step by step

Say what the question asks. Evaluate f(2) means the input is 2; find its output. It does not ask which input produces 2.
Copy the formula, replacing every input letter with the given input in parentheses.
For a number, do powers first, then multiplication, then addition and subtraction. For an expression, multiply out parentheses before collecting like terms.
Check by putting a numerical input into both the starting recipe and the expanded expression. Matching results can catch an expansion error.

Strategy

Evaluate the entire input

Is the input a number?

YesKeep it in parentheses and follow the order of operations.

NoKeep the whole letter expression in parentheses, then distribute and collect like terms.

↓

Does the input have a minus sign or more than one term?

YesKeep its parentheses through every power and multiplication.

NoStill replace every occurrence of the input variable.

Say what the question asks. Evaluate f(2) means the input is 2; find its output. It does not ask which input produces 2.
Copy the formula, replacing every input letter with the given input in parentheses.
For a number, do powers first, then multiplication, then addition and subtraction. For an expression, multiply out parentheses before collecting like terms.
Check by putting a numerical input into both the starting recipe and the expanded expression. Matching results can catch an expansion error.

Worked examplePositive and negative inputs

For f(x) = $x^{2}$ + 3x − 4, evaluate f(2) and f(−3). Both questions give an input and ask for its output.

Squaring the negative input gives 9, and three times that input gives −9.

What it asks. Use each stated input to find its output from the named formula.

Every input slot receives the complete input, including its sign.
Square a negative input in parentheses: (−2 $) squared$ = 4. A constant recipe gives the same output for every input.
Keep separate requested outputs on separate answer lines.

f(2) = (2 $) squared$ + 3(2) − 4.The input 2 replaces every x in the formula.
f(2) = 4 + 6 − 4 = 10 − 4 = 6.Do the square and multiplication before adding and subtracting.
f(−3) = (−3 $) squared$ + 3(−3) − 4.Parentheses keep the negative sign attached to the whole input.
f(−3) = 9 − 9 − 4 = 0 − 4 = −4.A negative times a negative is positive, but 3 times a negative is negative.

Answer

f(2) = 6.
f(−3) = −4.

Check At input 2, the pieces 4, 6, and −4 total 6. At input −3, the first two pieces are 9 and −9, so they cancel and leave −4.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Evaluation rung 1: a constant and a line

Let k(x) = 5 and L(x) = 4x − 3. Evaluate k(3) and L(4). Each question gives an input and asks for its output.

Keep the recipe's operations in their stated order.

What it asks. Use each stated input to find its output from the named formula.

Every input slot receives the complete input, including its sign.
Square a negative input in parentheses: (−2 $) squared$ = 4. A constant recipe gives the same output for every input.
Keep separate requested outputs on separate answer lines.

k(3) = 5.This formula always returns 5 and contains no input variable to replace.
L(4) = 4(4) − 3.Replace the x in the second formula with 4.
L(4) = 16 − 3 = 13.Multiply before subtracting.

Answer

k(3) = 5.
L(4) = 13.

Check The first rule returns 5 for every input. The second gives 4 × 4 − 3 = 13; adding 3 to that output and dividing by 4 recovers the input 4.

Rung 2Rung 2: protect a negative input

Let r(x) = 3x − 7. Evaluate r(−4). This means use −4 as the input and find the output.

Subtracting 7 from −12 lands at −19.

What it asks. Use each stated input to find its output from the named formula.

Every input slot receives the complete input, including its sign.
Square a negative input in parentheses: (−2 $) squared$ = 4. A constant recipe gives the same output for every input.
Keep separate requested outputs on separate answer lines.

r(−4) = 3(−4) − 7.The whole negative input replaces x.
r(−4) = −12 − 7.A positive times a negative is negative.
r(−4) = −19.Starting at −12 and subtracting 7 moves seven more places left.

Answer

r(−4) = −19

Check Undo the recipe on the output: −19 + 7 = −12, then −12 ÷ 3 = −4, the given input.

Rung 3Rung 3: a sum in both input spots

For s(x) = $x^{2}$ , evaluate s(a + h). The input is the entire sum a + h; find its output expression.

Every rectangle is one required product.

What it asks. Use the entire expression as input and find the simplified output expression.

Every input slot receives the complete input, including its sign.
A square of a sum includes two middle products. A negative sign outside parentheses changes every enclosed sign.
Keep separate requested outputs on separate answer lines.

s(a + h) = (a + h $) squared$ = (a + h)(a + h).A square multiplies the whole input by itself.
s(a + h) = $a^{2}$ + ah + ha + $h^{2}$ .Multiply each term in the first parentheses by each term in the second.
s(a + h) = $a^{2}$ + 2ah + $h^{2}$ .ah and ha are equal, so together they are 2ah.

Answer

s(a + h) = + 2ah +

Check Set a = 2 and h = 3. The input is 5 and its square is 25. The expansion gives 4 + 12 + 9 = 25.

Rung 4An expression input with a subtraction

For q(x) = $x^{2}$ − x, find q(a + 1). The whole input is a + 1; find the output expression.

q(a + 1) = (a + 1 − (a + 1).

(a + 1 = (a + 1)(a + 1) = + a + a + 1 = + 2a + 1.

q(a + 1) = + 2a + 1 − a − 1.

q(a + 1) = + a.

Read these successive steps along with their reasons.

What it asks. Use the entire expression as input and find the simplified output expression.

Every input slot receives the complete input, including its sign.
A square of a sum includes two middle products. A negative sign outside parentheses changes every enclosed sign.
Keep separate requested outputs on separate answer lines.

q(a + 1) = (a + 1 $) squared$ − (a + 1).Both copies of x receive the entire input a + 1.
(a + 1 $) squared$ = (a + 1)(a + 1) = $a^{2}$ + a + a + 1 = $a^{2}$ + 2a + 1.Multiply every term in the first parentheses by every term in the second.
q(a + 1) = $a^{2}$ + 2a + 1 − a − 1.Subtracting the whole input changes the signs of both a and 1.
q(a + 1) = $a^{2}$ + a.2a − a = a and 1 − 1 = 0.

Answer

q(a + 1) = + a

Check Choose a = 2. The input is 3, so the original formula gives q(3) = − 3 = 6. The new expression gives + 2 = 6 too.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: f(a + h) = f(a) + h.

The change happens to the input before the function recipe runs. The output generally changes by a different amount. Here f(3) = 14 while f(1) + 2 = 2.

✓ Instead: Replace every x by a + h: f(a + h) = (a + h + 3(a + h) − 4.

✗ Not this: f(−3) = − − 9 − 4 = −22.

− means negate the square of 3. The formula asks to square the entire input −3 instead.

✓ Instead: f(−3) = (−3 + 3(−3) − 4 = 9 − 9 − 4 = −4.

✗ Not this: (a + h = + .

This leaves out the products ah and ha. With a = 1 and h = 2, the left side is 9 while the proposed right side is 5.

✓ Instead: (a + h = + 2ah + , giving 1 + 4 + 4 = 9 in that check.

Tips and tricks

Put a seat belt of parentheses around each whole input before doing arithmetic.
A numerical spot-check can expose a wrong expression. A derivation proves the expression for every allowed input.

Trap. Losing the middle term when squaring a sum. Write (a + h)(a + h) and keep all four products before combining the two ah terms.