Pre-Calculus & Functions Review

01What a function actually is

A function is a rule that assigns to each input exactly one output. We write $f(x)$ to mean "the output of $f$ when the input is $x$." The set of allowed inputs is the domain; the set of resulting outputs is the range. The "exactly one output" part matters: $y = x^2$ is a function (each $x$ gives one $y$), but $x = y^2$ is not (each positive $x$ gives two $y$ values). The vertical line test captures this graphically — if any vertical line hits the graph more than once, it isn't a function.

Function notation is not multiplication. $f(3)$ means "evaluate $f$ at $3$," so if $f(x) = x^2 - 5$, then $f(3) = 9 - 5 = 4$. You'll also evaluate at expressions: $f(x+h) = (x+h)^2 - 5$. This exact move powers the limit definition of the derivative, so get comfortable now.

02Finding the domain

Three rules cover almost every Calc 1 case:

1. Denominators can't be zero. For $f(x) = \dfrac{1}{x-2}$, exclude $x = 2$.
2. Even roots need non-negative insides. For $g(x) = \sqrt{x-4}$, require $x - 4 \ge 0$, so $x \ge 4$.
3. Logarithms need positive insides. For $h(x) = \ln(x+1)$, require $x + 1 > 0$, so $x > -1$.

When a function combines these, you intersect the restrictions. Domain questions reappear constantly in limits and continuity, so treat them as reflexes.

03The function families you must know

Calc 1 recycles the same handful of families. You should recognize each shape instantly:

• Polynomials — $x^2$, $x^3 - 2x$. Domain is always all reals. Smooth, no breaks.
• Rational functions — ratios like $\dfrac{x+1}{x-3}$. Watch for vertical asymptotes where the denominator is zero.
• Exponentials — $2^x$, $e^x$. Grow or decay fast; always positive; horizontal asymptote at $y=0$.
• Logarithms — $\ln x$, $\log x$. Inverse of exponentials; defined only for positive inputs; vertical asymptote at $x=0$.
• Trigonometric — $\sin x$, $\cos x$, $\tan x$. Periodic and oscillating. $\sin$ and $\cos$ live between $-1$ and $1$.

04Combining and composing functions

You can add, subtract, multiply, and divide functions the obvious way: $(f+g)(x) = f(x) + g(x)$. More important for calculus is composition, written $(f \circ g)(x) = f(g(x))$ — you feed the output of $g$ into $f$. If $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$, then $f(g(x)) = \sqrt{x^2 + 1}$. Order matters: $g(f(x)) = (\sqrt{x})^2 + 1 = x + 1$ (for $x \ge 0$) is different. Recognizing a complicated function as a composition of simpler ones is exactly the skill the chain rule will demand later, so practice decomposing: $\sqrt{x^2+1}$ is "square root of (something)," where the something is $x^2 + 1$.

05Transformations

Starting from a base graph $y = f(x)$:

• $f(x) + c$ shifts up by $c$; $f(x) - c$ shifts down.
• $f(x - c)$ shifts right by $c$ (note the counterintuitive sign); $f(x+c)$ shifts left.
• $-f(x)$ flips vertically; $f(-x)$ flips horizontally.
• $a\,f(x)$ stretches vertically by factor $a$.

Knowing these lets you sketch $y = (x-2)^2 + 3$ without plotting points: it's the parabola shifted right 2 and up 3.

06Trigonometry survival kit

You don't need all of trig, but you need this cold. On the unit circle, $\cos\theta$ is the $x$-coordinate and $\sin\theta$ is the $y$-coordinate. Memorize the key angles:

$$\sin 0 = 0,\quad \sin\tfrac{\pi}{6} = \tfrac{1}{2},\quad \sin\tfrac{\pi}{4} = \tfrac{\sqrt2}{2},\quad \sin\tfrac{\pi}{3} = \tfrac{\sqrt3}{2},\quad \sin\tfrac{\pi}{2} = 1$$

Cosine runs the same values in reverse. The identity you'll use most is the Pythagorean identity:

$$\sin^2\theta + \cos^2\theta = 1$$

Also know $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ and that angles are measured in radians in calculus ($180° = \pi$ radians), not degrees. This is non-negotiable: the derivative formulas for trig functions only work in radians.

07Exponent and logarithm rules

These algebra rules get used silently in nearly every derivative and integral:

$$x^a \cdot x^b = x^{a+b}, \qquad \frac{x^a}{x^b} = x^{a-b}, \qquad (x^a)^b = x^{ab}, \qquad x^{-a} = \frac{1}{x^a}, \qquad x^{1/n} = \sqrt[n]{x}$$

For logs (where $\ln$ is log base $e$):

$$\ln(ab) = \ln a + \ln b, \qquad \ln\frac{a}{b} = \ln a - \ln b, \qquad \ln(a^b) = b\ln a, \qquad \ln e^x = x$$

Rewriting $\dfrac{1}{\sqrt{x}}$ as $x^{-1/2}$ is the kind of move that makes derivatives trivial instead of painful.

08Factoring — the skill that saves your limits

When you plug a value into a limit and get $\frac{0}{0}$, the fix is almost always factoring and canceling. Keep these sharp: difference of squares $a^2 - b^2 = (a-b)(a+b)$; the quadratic factor patterns; and the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For example, $\dfrac{x^2 - 9}{x - 3} = \dfrac{(x-3)(x+3)}{x-3} = x + 3$ once you cancel. That single technique resolves a huge fraction of Calc 1 limit problems.