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Evaluating Functions: f(x) Notation

Read function notation and evaluate linear and quadratic functions for any input — whole numbers, negatives, decimals, or values read from a graph or table.

Practice this skill 25 min Evaluate linear and quadratic functions Leer en español
Key idea

What you'll learn

In this lesson you will learn to read function notation, like f(x) = 3x + 4, and to evaluate a function — find its output for a given input. You will work with linear functions (f(x) = mx + b) and quadratic functions (f(x) = ax^2 + bx + c), including tricky inputs such as negative numbers and decimals, and you will learn to work backwards from an output to the input.

Why it matters on the GED. Function notation appears all over the math test:

  • "If f(x) = x^2 - 3x + 2, what is f(-2)?"
  • "A plumber charges C(h) = 65h + 45 dollars for h hours. What is the charge for 3.5 hours?"
  • "The height of a ball is h(t) = -16t^2 + 48t + 6. What is the height after 2 seconds?"
  • "Based on the graph of f, what is f(4)?"

Why it matters in real life. Phone bills, taxi fares, and paycheck calculators are all functions: you put in a number (minutes, miles, hours) and a rule gives you exactly one number back (the cost). Evaluating a function is just "plugging in" carefully.

Words to know

  • Function — a rule that gives exactly one output for each input.
  • Function notation f(x) — read "f of x." It means "the output of function f when the input is x." It does not mean f times x.
  • Input — the number inside the parentheses. In f(5), the input is 5.
  • Output — the value of f(x). If f(5) = 19, the output is 19, and the point (5, 19) is on the graph.
  • Evaluate — replace x with a number and simplify. Evaluate f(x) = 2x + 1 at x = 3: f(3) = 2(3) + 1 = 7.
  • Substitute — put a number in place of a variable, inside parentheses: x = -2 becomes (-2).
  • Linear function — f(x) = mx + b; its graph is a straight line. Example: f(x) = 4x - 9.
  • Quadratic function — f(x) = ax^2 + bx + c with a \ne 0; its graph is a parabola. Example: g(x) = x^2 - 5x + 6.
  • Order of operations — Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

All Math words in the glossary

1. Reading function notation

A function is like a machine: an input goes in, the rule acts on it, and an output comes out.

\underbrace{f}_{\text{name}}(\underbrace{x}_{\text{input}}) = \underbrace{3x + 4}_{\text{rule}}
  • f(5) means "put 5 into f." So f(5) = 3(5) + 4 = 19.
  • The letter does not matter: g(x), h(t), C(n), P(x) all work the same way. Often the letter hints at the meaning — C for cost, h for height, t for time.
  • f(5) = 19 is a point on the graph: (5, 19). The input is the x-coordinate and the output is the y-coordinate. So y = f(x).
A function machine. The input 5 goes into a box labelled f(x) = 3x + 4, where the work 3(5) + 4 = 15 + 4 is shown, and the output 19 comes out. Caption: f(5) = 19 means input 5 gives output 19, which is the point (5, 19) on the graph. f(5) is not f times 5.input5f(x) = 3x + 43(5) + 4 = 15 + 4output19f(5) = 19 means input 5 → output 19 → point (5, 19) on the graphf(5) is NOT “f times 5”

Important: f(x) is not multiplication. f(5) is not "f \times 5."

Two or more functions. If f(x) = x^2 + 1 and g(x) = 2x - 3, then

f(2) - g(-1) = \big(2^2 + 1\big) - \big(2(-1) - 3\big) = 5 - (-5) = 10.

Evaluate each piece separately, then combine.

f(0) is special: it is the y-intercept of the graph. For f(x) = 3x + 4, f(0) = 4, so the graph crosses the y-axis at (0, 4).

2. Evaluating safely: parentheses, exponents, negatives

Step-by-step method

  1. Rewrite the rule with empty parentheses wherever x appears: f(\ ) = (\ )^2 - 3(\ ) + 2.
  2. Put the input inside every pair of parentheses.
  3. Simplify using the order of operations: exponents first, then multiplication, then addition and subtraction from left to right.

Example — linear: f(x) = 10 - 2x, find f(-3).

f(-3) = 10 - 2(-3) = 10 + 6 = 16

Example — quadratic: f(x) = x^2 - 3x + 2, find f(-2).

f(-2) = (-2)^2 - 3(-2) + 2 = 4 + 6 + 2 = 12
Two function machines that use empty parentheses. First: input −3 goes into f(x) = 10 − 2( ); the work 10 − 2(−3) = 10 + 6 gives output 16. Second: input −2 goes into f(x) = ( )² − 3( ) + 2; the work (−2)² − 3(−2) + 2 = 4 + 6 + 2 gives output 12. Caption: put the input inside every pair of parentheses, then do exponents, multiplication, then addition and subtraction.input−3f(x) = 10 − 2( )10 − 2(−3) = 10 + 6output16input−2f(x) = ( )² − 3( ) + 2(−2)² − 3(−2) + 2 = 4 + 6 + 2output12Put the input inside EVERY pair of parentheses, then: exponents → multiply → add/subtract

The negative-square trap. These are different:

ExpressionMeaningValue
(-3)^2(-3)(-3)9
-3^2-(3 \cdot 3)-9
-x^2 when x = -3-\big((-3)^2\big)-9
2x^2 when x = 32 \cdot 3^2 = 2 \cdot 918 (not 6^2 = 36)

A squared number is never negative, so (-2)^2 = 4. But a minus sign in front of the squared term stays: in h(t) = -16t^2 + \dots, the term -16t^2 is always zero or negative.

Decimals and fractions work the same way: C(h) = 65h + 45 at h = 3.5 gives 65(3.5) + 45 = 227.5 + 45 = 272.5. At x = \tfrac{1}{2}: f(x) = 4x^2 gives 4\left(\tfrac{1}{2}\right)^2 = 4 \cdot \tfrac{1}{4} = 1.

3. Values from graphs and tables — and working backwards

From a graph: f(a) is the y-value of the point on the graph directly above or below x = a.

Graph of y = f(x), an upward-opening parabola on a coordinate plane from -4 to 6 horizontally and -5 to 6 vertically. Marked points on the curve: (-2, 5), (-1, 0), (0, -3), (1, -4) which is the lowest point, (2, -3), (3, 0), and (4, 5).−4−3−2−1123456−5−4−3−2−1123456xyy = f(x)

On this graph: f(4) = 5 (go to x = 4, up to the curve, read y = 5); f(0) = -3; f(1) = -4 (the lowest point); and f(-1) = 0. (Its rule is f(x) = x^2 - 2x - 3; for example f(4) = 16 - 8 - 3 = 5 ✓.)

Do not confuse f(3) — the output when x = 3, here 0 — with "where does f(x) = 3?", which asks for an input.

From a table: find the input in the x row; the output is directly below it.

x-1025
g(x)74-2-11

So g(2) = -2 and g(-1) = 7.

Working backwards (solving). "For what x is f(x) = 25?" gives you the output; you find the input by solving an equation.

  • Linear: f(x) = 4x - 7 and f(x) = 25 → 4x - 7 = 25 → 4x = 32 → x = 8.
  • Quadratic: p(x) = x^2 - 6 and p(x) = 30 → x^2 = 36 → x = 6 or x = -6. A quadratic can have two inputs with the same output — remember the negative root.
Tip

What the answer means in a word problem

On the GED, a function value often comes with a story, and some questions ask what a statement means rather than asking you to compute it.

  • C(3.5) = 272.5 for a plumber's bill means "a job that takes 3.5 hours costs $272.50." The number in the parentheses is the input (hours); the number after the equals sign is the output (dollars).
  • h(2) = 38 for a ball means "after 2 seconds, the ball is 38 feet high." It does not mean the ball travels 2 feet in 38 seconds.
  • C(0) = 45 means "a zero-hour visit still costs $45" — the flat fee, which is also the y-intercept.

Making a table of values. To see how a function behaves, evaluate it at several inputs and list the results. For C(h) = 65h + 45:

h (hours)0123
C(h) (dollars)45110175240

The same table drawn as a graph:

Table of C(h) = 65h + 45 beside its graph. Table: h = 0, 1, 2, 3 hours give C = 45, 110, 175, 240 dollars, with +65 marked between rows. Graph: cost in dollars against hours, a straight line starting at the flat fee (0, 45) and passing through (1, 110), (2, 175), (3, 240). A red point marks (3.5, 272.50), the cost of a 3.5-hour job.C(h) = 65h + 45h (hours)C(h) ($)045111021753240+65+65+6501234050100150200250300Hours (h)Cost in dollars(0, 45)(1, 110)(2, 175)(3, 240)(3.5, 272.50)C(0) = 45: flat fee

Each extra hour adds $65, the rate of change. A table like this is also a handy way to check an answer choice quickly.

Formula

Forms to recognize

f(x) = mx + b \qquad\qquad f(x) = ax^2 + bx + c
  • x — the input; f(x) — the output (the same as y)
  • m — slope (rate of change) of a linear function; b — its y-intercept, which equals f(0)
  • a, b, c — the coefficients of a quadratic; c = f(0), the y-intercept

Formula sheet: the slope-intercept form y = mx + b, the standard form of a quadratic y = ax^2 + bx + c, and the quadratic formula are on the GED formula sheet. The meaning of function notation f(x) and the rule "substitute, then follow the order of operations" are not on the sheet — you must know them.

Worked example

Example 1 (easy): a linear function

Let f(x) = 4x - 9. Find f(5) and f(-2).

  1. f(5) = 4(5) - 9 = 20 - 9 = 11.
  2. f(-2) = 4(-2) - 9 = -8 - 9 = -17.
Graph of the line f(x) = 4x − 9. Marked points: (−2, −17), (0, −9) and (5, 11), so f(−2) = −17 and f(5) = 11. A green dashed step from (−2, −17) runs 7 to the right and rises 28 up to (5, 11), showing rise = 4 × 7 = 28.−3−2−1123456−20−15−10−5510150xyf(5) = 11 → (5, 11)(−2, −17)run = 7rise = 4 × 7 = 28(0, −9)

Answer: f(5) = \mathbf{11} and f(-2) = \mathbf{-17}.

Check: The slope is 4, so going from x = -2 to x = 5 (7 steps) should raise the output by 4 \times 7 = 28: -17 + 28 = 11 ✓

Worked example

Example 2: a quadratic with a negative input

Let g(x) = x^2 - 5x + 6. Find g(-3).

  1. Substitute with parentheses: g(-3) = (-3)^2 - 5(-3) + 6.
  2. Exponent first: (-3)^2 = 9.
  3. Multiply: -5(-3) = +15.
  4. Add: 9 + 15 + 6 = 30.
Graph of the parabola g(x) = x² − 5x + 6, opening upward, crossing the x-axis at (2, 0) and (3, 0) and the y-axis at (0, 6). A red point marks (−3, 30), with dashed guides to the axes. Beside it: g(−3) = (−3)² − 5(−3) + 6 = 9 + 15 + 6 = 30.−4−3−2−1123456−551015202530350xy(−3, 30)(0, 6)(2, 0)(3, 0)g(−3) = (−3)² − 5(−3) + 6= 9 + 15 + 6= 30

Answer: g(-3) = \mathbf{30}.

Check: Factor g(x) = (x - 2)(x - 3). Then g(-3) = (-3 - 2)(-3 - 3) = (-5)(-6) = 30 ✓

Worked example

Example 3: real-world linear function (taxi fare)

A taxi company charges C(m) = 3.25 + 2.40m dollars for a ride of m miles. How much does a 12.5-mile ride cost?

  1. Substitute m = 12.5: C(12.5) = 3.25 + 2.40(12.5).
  2. Multiply first: 2.40 \times 12.5 = 30.
  3. Add: 3.25 + 30 = 33.25.
Graph of the taxi fare C(m) = 3.25 + 2.40m, fare in dollars against miles from 0 to 14. The line starts at the base fare (0, 3.25) and rises 2.40 dollars per mile. A red point at (12.5, 33.25) with dashed guides shows that a 12.5-mile ride costs 33.25 dollars.024681012140510152025303540Miles (m)Fare in dollars(12.5, 33.25)(0, 3.25): base fareC(m) = 3.25 + 2.40m

Answer: $33.25

Check: Estimate: about $2.40 per mile × 12 miles ≈ $29, plus about $3 → about $32–$33 ✓

Worked example

Example 4: real-world quadratic (height of a ball)

A ball is thrown upward. Its height in feet after t seconds is h(t) = -16t^2 + 64t + 5. How high is the ball after 1.5 seconds?

  1. Substitute: h(1.5) = -16(1.5)^2 + 64(1.5) + 5.
  2. Exponent: (1.5)^2 = 2.25.
  3. Multiply: -16(2.25) = -36 and 64(1.5) = 96.
  4. Add: -36 + 96 + 5 = 65.
Graph of the ball height h(t) = −16t² + 64t + 5, height in feet against time in seconds. The curve starts at (0, 5), rises through (1, 53), the red point (1.5, 65) and (2, 69), then falls back to the ground a little after 4 seconds. A dashed line marks t = 1.5.0123401020304050607080Time in seconds (t)Height in feet(0, 5)(1, 53)(1.5, 65)(2, 69)h(t) = −16t² + 64t + 5

Answer: 65 feet

Check: h(1) = -16 + 64 + 5 = 53 and h(2) = -64 + 128 + 5 = 69. A value of 65 between them makes sense as the ball rises ✓

Worked example

Example 5 (GED-hard): work backwards

(a) f(x) = 3x + 7. For what value a is f(a) = 31? (b) p(x) = 2x^2 - 8. For which values of x is p(x) = 42?

(a)

  1. Set the rule equal to the output: 3a + 7 = 31.
  2. Subtract 7: 3a = 24.
  3. Divide by 3: a = 8.

(b)

  1. Set up: 2x^2 - 8 = 42.
  2. Add 8: 2x^2 = 50. Divide by 2: x^2 = 25.
  3. Take the square root — both signs: x = 5 or x = -5.
Working backwards from an output, in two graphs. (a) The line f(x) = 3x + 7 meets the dashed horizontal line y = 31 at (8, 31); a green arrow drops to the x-axis at a = 8. (b) The parabola p(x) = 2x² − 8, with lowest point (0, −8), meets the dashed line y = 42 at two points, (−5, 42) and (5, 42); green arrows drop to x = −5 and x = 5, showing a quadratic can have two inputs with the same output.(a) f(x) = 3x + 7, output 310246810010203040xy(8, 31)y = 31a = 8(b) p(x) = 2x² − 8, output 42−6−4−224620400xy(−5, 42)(5, 42)x = −5x = 5(0, −8)

Answer: (a) a = \mathbf{8}; (b) x = \mathbf{5} or x = \mathbf{-5}.

Check: f(8) = 24 + 7 = 31 ✓; p(5) = 2(25) - 8 = 42 ✓ and p(-5) = 2(25) - 8 = 42 ✓

Calculator tip

TI-30XS MultiView tips

  • Always type negative inputs with the (−) key, inside parentheses, then square: ( (−) 3 ) x² gives 9. Typing (−) 3 x² without parentheses gives −9, because the calculator squares 3 first.
  • Powers: use x² for squares and ^ for other powers.
  • Whole expressions at once: for h(1.5) = -16(1.5)^2 + 64(1.5) + 5, type (−) 16 ( 1.5 ) x² + 64 ( 1.5 ) + 5 enter → 65.
  • Fractions: use n/d for inputs like \tfrac{1}{2}, and toggle to switch the answer between fraction and decimal.
  • When NOT to use it: for small whole numbers like f(5) = 3(5) + 4, mental math is faster and avoids typing errors. Many simple evaluations can appear in the calculator-free part of the test.
Common mistake

Common mistakes

  • Squaring a negative without parentheses. ✗ (-2)^2 - 3(-2) + 2 evaluated as -4 + 6 + 2 = 4. ✓ (-2)^2 = 4, so the answer is 4 + 6 + 2 = 12.
  • Losing the sign in -bx. ✗ -3(-2) = -6. ✓ Negative × negative = positive: -3(-2) = +6.
  • Squaring the coefficient too. ✗ 2x^2 at x = 3 as (2 \cdot 3)^2 = 36. ✓ Only x is squared: 2 \cdot 3^2 = 2 \cdot 9 = 18.
  • Forgetting the square. ✗ -16t^2 at t = 2 as -16 \cdot 2 = -32. ✓ -16 \cdot 2^2 = -16 \cdot 4 = -64.
  • Treating f(x) as multiplication, or answering the input instead of the output. f(3) asks for a y-value, not "f \times 3."
  • Forgetting the negative solution when working backwards with a quadratic. ✗ x^2 = 36 \Rightarrow x = 6 only. ✓ x = 6 or x = -6.
Test strategy

GED test strategy

  • Write the rule with parentheses on your erasable note board first, then fill in the number. This one habit prevents most sign errors.
  • Read the question twice: "find f(4)" (plug in 4) is different from "find x when f(x) = 4" (solve).
  • Eliminate answer choices that come from the classic slips: a choice that equals your answer with a sign flipped, or one that equals (\text{coefficient} \times x)^2.
  • Plug answers back in for "work backwards" questions: if the choices are 6 and -6, test them in the function.
  • Estimate real-world answers: a taxi ride shouldn't cost $330 for 12 miles; a ball thrown by hand shouldn't be 1,000 feet high.
  • Units: in word problems, attach the unit (dollars, feet, seconds) to your answer and make sure it matches the question.

Check yourself

Evaluate linear and quadratic functionsNo calculator

If f(x) = 3x + 4, what is the value of f(5)?

Answer choices
Hint

Replace x with 5. Remember that 3x means 3 times x.

Check yourself

Evaluate linear and quadratic functionsNo calculatorFill in

If g(x) = x^2 + 1, what is g(4)?

g(4) =

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Replace x with 4. Square first, then add.

Summary

  • f(x) means "the output of f for input x" — it is not multiplication; f(a) = b means the point (a, b) is on the graph.
  • To evaluate, substitute inside parentheses, then use the order of operations: exponents → multiply → add/subtract.
  • (-3)^2 = 9, but -3^2 = -9; in ax^2 only x is squared.
  • From a graph or table, f(a) is the y-value that goes with x = a; f(0) is the y-intercept.
  • Working backwards: set f(x) equal to the given output and solve — a quadratic can give two answers.
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