Functions

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

1. Do compositions from the inside out. For inverses, solve $f(a)=b$ instead of finding a formula for $f^{-1}$.
2. For a functional equation, substitute a specific value, then its partner value, and solve the system you get.
3. 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.
4. To move a graph, follow two or three landmark points, such as the intercepts or the vertex.
5. 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)