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Functions

Function evaluation, composition, inverses, functional equations, graphs and transformations

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FunctionsAMC 10 Step by Step

Name: Date: October 28, 2026

all 25 problems, easiest first

What you need to know

Composition. $f(g(x))$ means: apply $g$ first, then apply $f$ to the result. Work from the inside out. Order matters: $f(g(x))$ and $g(f(x))$ are usually different. Example: if $f(x)=x+1$ and $g(x)=x^2$ , then $f(g(3))=f(9)=10$ but $g(f(3))=g(4)=16$ .
Inverse. $f^{-1}(b)=a$ means exactly the same as $f(a)=b$ , so a question about $f^{-1}$ is really "solve $f(a)=b$ ". The graph of $f^{-1}$ is the graph of $f$ reflected across the line $y=x$ . Example: if $f(x)=2x+3$ , then $f^{-1}(11)$ is the $a$ with $2a+3=11$ , so $f^{-1}(11)=4$ .
Moving graphs. Let $c$ be a positive number. $f(x)+c$ moves the graph up by $c$ , and $f(x-c)$ moves it right by $c$ . $f(-x)$ flips it across the $y$ -axis, and $-f(x)$ flips it across the $x$ -axis. $f(cx)$ squeezes it toward the $y$ -axis by a factor of $c$ (when $c>1$ ). Example: $y=(x-2)^2+1$ is $y=x^2$ moved right $2$ and up $1$ , so its lowest point is $(2,1)$ .
Functional equations. These give a rule for $f$ instead of a formula. Solve them by substitution: plug in $0$ , $1$ , $-1$ , and the "partner" input that appears in the equation (such as $\frac1x$ , $1-x$ , or $-x$ ). This gives a small system of linear equations in the unknown values. Example: if $f(x)-2f(-x)=x$ for all $x$ , then $x=0$ gives $f(0)-2f(0)=0$ , so $f(0)=0$ .
Symmetry. If $f(x)=f(2a-x)$ for every $x$ , the graph is symmetric about the vertical line $x=a$ , because $x$ and $2a-x$ are the same distance from $a$ on opposite sides. Example: $f(x)=(x-3)^2$ has $f(x)=f(6-x)$ (here $a=3$ ), and indeed $f(1)=f(5)=4$ .
Piecewise and floor rules. If $f$ has different rules for different inputs, or uses the floor $\lfloor x\rfloor$ (the greatest integer less than or equal to $x$ ), split into cases by the input. Example: $\lfloor2.7\rfloor=2$ but $\lfloor-2.7\rfloor=-3$ .

How AMC 10 tests it

Problems 3–10: evaluate a composition or an inverse from a formula or a small table, often working one step backwards.
Problems 8–15: " $f(x)+2f\!\left(\frac1x\right)=3x$ ; find $f(2)$ ". Substitute $x=2$ and $x=\frac12$ and solve the two equations in two unknowns.
Problems 10–18: rules such as $f(x+y)=f(x)f(y)$ , where a clever choice of inputs pins down the value.
Graph questions: pick out the graph of $y=f(x-1)+2$ , or count the solutions of $f(f(x))=c$ from a sketch.
Problems 15–22: a function applied over and over, written $f^{(n)}(x)$ for $n$ applications (find when it starts repeating), or counting fixed points (inputs with $f(x)=x$ ).

Standard approaches

Do compositions from the inside out. For inverses, solve $f(a)=b$ instead of finding a formula for $f^{-1}$ .
For a functional equation, substitute a specific value, then its partner value, and solve the system you get.
For a function applied many times, work out $f(f(x))$ and $f(f(f(x)))$ with algebra and watch for $x$ to come back. Example: for $f(x)=\frac1{1-x}$ , $f(f(x))=\frac{x-1}x$ and $f(f(f(x)))=x$ , so the values repeat every $3$ steps.
To move a graph, follow two or three landmark points, such as the intercepts or the vertex.
To count solutions from a graph, count crossings with horizontal lines, one layer at a time. Example: for $f(f(x))=c$ , first find every $y$ with $f(y)=c$ , then count the solutions of $f(x)=y$ for each one.

Worked example

The function $f$ satisfies $f(x)+2f(1-x)=3x^2$ for all real $x$ . What is $f(3)$ ?

(A) $-3$ (B) $-1$ (C) $1$ (D) $3$ (E) $5$

Substitute $x=3$ . Since $1-3=-2$ , this gives $f(3)+2f(-2)=27$ .

The equation pairs $x$ with $1-x$ , so the partner of $3$ is $-2$ . Substitute $x=-2$ . Since $1-(-2)=3$ and $3(-2)^2=12$ , this gives $f(-2)+2f(3)=12$ , so $f(-2)=12-2f(3)$ .

Put that into the first equation:
$f(3)+2\bigl(12-2f(3)\bigr)=27\quad\Longrightarrow\quad 24-3f(3)=27\quad\Longrightarrow\quad f(3)=-1.$
Check: $f(x)=2(1-x)^2-x^2$ satisfies the equation, and $f(3)=2\cdot4-9=-1$ .

The answer is $\boxed{\textbf{(B)}\ -1}$ .

Pitfalls

Composing in the wrong order ( $f(g(x))$ versus $g(f(x))$ ).
Confusing $f^{-1}(x)$ with $\frac1{f(x)}$ . Example: for $f(x)=2x$ , $f^{-1}(x)=\frac x2$ , but $\frac1{f(x)}=\frac1{2x}$ .
Shifting the wrong way: $f(x-2)$ moves the graph right, not left. Example: $y=(x-2)^2$ has its vertex at $x=2$ .
Using only one substituted equation. You need both the original and the partner equation to get rid of the extra unknown value.

Traps that recur

Bounding |f(800) − f(400)| by ½·400 = 200 directly (ignoring f(300) = f(900)) and answering 100. (2022 AMC 10B #24)
Applying the rule as if diamond were multiplication or subtraction, or computing 6 diamond x = 6x instead of 6/x. (2016 AMC 10A #23)
Summing 1 through 100 instead of 1 through 99 (exponent 5050), or reading f(2n) = n f(n) as f(2^n) = 2^n f(2^(n-1)). (2004 AMC 10A #24)
Plugging 3z in for x on the right side (as if f(x) = x^2 + x + 1), instead of first finding f(t) for a general input t. (2000 AMC 10 #24)

1.2022 AMC 10B #1
Define $x\diamond y$ to be $|x-y|$ for all real numbers $x$ and $y.$ What is the value of $(1\diamond(2\diamond3))-((1\diamond2)\diamond3)?$
1. A) ${-}2$
2. B) ${-}1$
3. C) $0$
4. D) $1$
5. E) $2$
2.2016 AMC 10B #2
If $n\heartsuit m=n^3m^2$ , what is $\frac{2\heartsuit 4}{4\heartsuit 2}$ ?
1. A) $\frac{1}{4}$
2. B) $\frac{1}{2}$
3. C) $1$
4. D) $2$
5. E) $4$
3.2015 AMC 10B #7
Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$ . What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$ ?
1. A) $-\dfrac{7}{30}$
2. B) $-\dfrac{1}{6}$
3. C) $0$
4. D) $\dfrac{1}{6}$
5. E) $\dfrac{7}{30}$
4.2011 AMC 10A #3
Suppose $[a\ b]$ denotes the average of $a$ and $b$ , and $\{a\ b\ c\}$ denotes the average of $a, b$ , and $c$ . What is $\{\{1\ 1\ 0\}\ [0\ 1]\ 0\}$ ?
1. A) $\frac{2}{9}$
2. B) $\frac{5}{18}$
3. C) $\frac{1}{3}$
4. D) $\frac{7}{18}$
5. E) $\frac{2}{3}$
5.2010 AMC 10B #4
For a real number $x$ , define $\heartsuit(x)$ to be the average of $x$ and $x^2$ . What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$ ?
1. A) $3$
2. B) $6$
3. C) $10$
4. D) $12$
5. E) $20$
6.2010 AMC 10A #6
For positive numbers $x$ and $y$ the operation $x\spadesuit y$ is defined as
$x\spadesuit y = x -\dfrac{1}{y}$
What is $2\spadesuit (2\spadesuit 2)$ ?
1. A) $\dfrac{2}{3}$
2. B) $1$
3. C) $\dfrac{4}{3}$
4. D) $\dfrac{5}{3}$
5. E) $2$
7.2008 AMC 10B #5
For real numbers $a$ and $b$ , define $a \$b$ $=(a-b)^2$ . What is $(x-y)^2\$(y-x)^2$ ?
1. A) $0$
2. B) $x^2+y^2$
3. C) $2x^2$
4. D) $2y^2$
5. E) $4xy$
8.2007 AMC 10A #2
Define $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$ . What is $\frac {6@2}{6\#2}$ ?
1. A) $- \frac {1}{2}$
2. B) $- \frac {1}{4}$
3. C) $\frac {1}{8}$
4. D) $\frac {1}{4}$
5. E) $\frac {1}{2}$
9.2007 AMC 10B #2
Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$
1. A) $-16$
2. B) $-8$
3. C) $0$
4. D) $8$
5. E) $16$
10.2006 AMC 10B #2
For real numbers $x$ and $y$ , define $x \spadesuit y = (x+y)(x-y)$ . What is $3 \spadesuit (4 \spadesuit 5)$ ?
1. A) $-72$
2. B) $-27$
3. C) $-24$
4. D) $24$
5. E) $72$
11.2006 AMC 10A #2
Define $x\otimes y=x^3-y$ . What is $h\otimes (h\otimes h)$ ?
1. A) $-h$
2. B) $0$
3. C) $h$
4. D) $2h$
5. E) $h^3$
12.2005 AMC 10B #4
For real numbers $a$ and $b$ , define $a \diamond b = \sqrt{a^2 + b^2}$ . What is the value of
$(5 \diamond 12) \diamond ((-12) \diamond (-5))$ ?
1. A) $0$
2. B) $\frac{17}{2}$
3. C) $13$
4. D) $13\sqrt{2}$
5. E) $26$
13.2005 AMC 10A #2
For each pair of real numbers $a \neq b$ , define the operation $\star$ as
$(a \star b) = \frac{a+b}{a-b}.$
What is the value of $((1 \star 2) \star 3)$ ?
1. A) $-\frac{2}{3}$
2. B) $-\frac{1}{5}$
3. C) $0$
4. D) $\frac{1}{2}$
5. E) $\text{This value is not defined.}$
14.2004 AMC 10A #2
For any three real numbers $a$ , $b$ , and $c$ , with $b\neq c$ , the operation $\otimes$ is defined by: $\otimes(a,b,c)=\frac{a}{b-c}$ What is $\otimes(\otimes(1,2,3),\otimes(2,3,1),\otimes(3,1,2))$ ?
1. A) $-\frac{1}{2}$
2. B) $-\frac{1}{4}$
3. C) $0$
4. D) $\frac{1}{4}$
5. E) $\frac{1}{2}$
15.2002 AMC 10A #2
Given that a, b, and c are non-zero real numbers, define $(a, b, c) = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ , find $(2, 12, 9)$ .
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
16.2002 AMC 10B #2
For the nonzero numbers a, b, and c, define
$D(a,b,c)=\frac{abc}{a+b+c}$
Find $D(2,4,6)$ .
1. A) $1$
2. B) $2$
3. C) $4$
4. D) $6$
5. E) $24$
17.2019 AMC 10B #9
The function $f$ is defined by $f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|$ for all real numbers $x$ , where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$ . What is the range of $f$ ?
1. A) $\{-1, 0\}$
2. B) $\text{The set of nonpositive integers}$
3. C) $\{-1, 0, 1\}$
4. D) $\{0\}$
5. E) $\text{The set of nonnegative integers}$
18.2011 AMC 10B #15
Let $@$ denote the "averaged with" operation: $a @ b = (a+b)/2$ . Which of the following distributive laws hold for all numbers $x, y,$ and $z$ ? $\text{I. x @ (y + z) = (x @ y) + (x @ z)}$ $\text{II. x + (y @ z) = (x + y) @ (x + z)}$ $\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}$
1. A) $\text{I only}$
2. B) $\text{II only}$
3. C) $\text{III only}$
4. D) $\text{I and III only}$
5. E) $\text{II and III only}$
19.2025 AMC 10A #9
Let $f(x) = 100x^3 - 300x^2 + 200x$ . For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$ ?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $\text{more than }4$
20.2021 AMC Fall 10A #16
The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$ .)
1. A) $\text{the }y\text{-axis}$
2. B) $\text{the line }x = 1$
3. C) $\text{the origin}$
4. D) $\text{ the point }\left(\dfrac12, 0\right)$
5. E) $\text{the point }(1,0)$
21.2021 AMC 10A #18
Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a\cdot b)=f(a)+f(b)$ for all positive rational numbers $a$ and $b$ . Furthermore, suppose that $f$ also has the property that $f(p)=p$ for every prime number $p$ . For which of the following numbers $x$ is $f(x)<0$ ?
1. A) $\frac{17}{32}$
2. B) $\frac{11}{16}$
3. C) $\frac{7}{9}$
4. D) $\frac{7}{6}$
5. E) $\frac{25}{11}$
22.2016 AMC 10A #23
A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$ . (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $p+q?$
1. A) $109$
2. B) $201$
3. C) $301$
4. D) $3049$
5. E) $33,601$
23.2004 AMC 10A #24
Let $f$ be a function with the following properties:
(i) $f(1) = 1$ , and
(ii) $f(2n) = n \cdot f(n)$ for any positive integer $n$ .
What is the value of $f(2^{100})$ ?
1. A) $1$
2. B) $2^{99}$
3. C) $2^{100}$
4. D) $2^{4950}$
5. E) $2^{9999}$
24.2000 AMC 10 #24
Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$ . Find the sum of all values of $z$ for which $f(3z) = 7$ .
1. A) $-1/3$
2. B) $-1/9$
3. C) $0$
4. D) $5/9$
5. E) $5/3$
25.2022 AMC 10B #24
Consider functions $f$ that satisfy $|f(x)-f(y)|\leq \frac{1}{2}|x-y|$ for all real numbers $x$ and $y$ . Of all such functions that also satisfy the equation $f(300) = f(900)$ , what is the greatest possible value of $f(f(800))-f(f(400))?$
1. A) $25$
2. B) $50$
3. C) $100$
4. D) $150$
5. E) $200$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.