Revision sheet

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 $f(x) \cdot g(x)$.

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.

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 $\dfrac{1}{f(x)}$.

Asymptotes after shifts

• $y = a \cdot b^{x} + k$: horizontal asymptote $y = k$.
• $y = \log_b(x - h) + k$: vertical asymptote $x = h$, domain $x > h$.