Revision sheet

Function notation

• $f(a)$ is the output for the input $a$.
• Replace every $x$ by $(a)$, with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$.
• On a graph: the height above $x = a$.

Watch out: $f(a + b)$ is not $f(a) + f(b)$.

Piecewise functions

1. Find the piece whose condition the input satisfies.
2. Use that piece's formula.
3. $|x| = x$ for $x \ge 0$, and $-x$ for $x < 0$.

Watch out: at a boundary, read $<$ and $\ge$ carefully.

Shifts and reflections

• $f(x - h)$: $h$ units right. $f(x) + k$: $k$ units up.
• $-f(x)$: flip across the $x$-axis. $f(-x)$: flip across the $y$-axis.
• Inside $f(\dots)$ acts on $x$; outside acts on $y$.

Watch out: $f(x + 3)$ moves the graph left.

Stretches, even and odd

• $a f(x)$: vertical stretch by $a$. $f(bx)$: horizontal compression by $b$.
• Even: $f(-x) = f(x)$, mirror image in the $y$-axis.
• Odd: $f(-x) = -f(x)$, symmetric about the origin.

Watch out: compute $f(-x)$; the signs of the coefficients do not decide.

Composition

• $f(g(x))$: apply $g$ first, then $f$.
• Work from the inside out.
• The order matters: $f(g(x))$ and $g(f(x))$ usually differ.

Watch out: $f(g(x))$ is not $f(x) \cdot g(x)$.

Inverse functions

1. Write $y = f(x)$.
2. Solve for $x$, undoing the last operation first.
3. Rename: $f^{-1}(x)$. Check that $f(f^{-1}(x)) = x$.

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$.