Precalculus course

1. 1.Domain and Range

   The domain of a function is the set of inputs it accepts; the range is the set of outputs it actually produces. When a function is given by a formula with no stated restriction, the domain is every real number that does not break the formula.

   • Denominator: exclude values that make it zero.
   • Square root: the inside must be $\ge 0$.
   • Logarithm: the inside must be $> 0$.
   Quick check: What is the smallest $x$ in the domain of $\sqrt{2x - 8}$?

   $4$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study domain and range

2. 2.Composition of Functions

   Composing functions means feeding the output of one into another: $(f \circ g)(x) = f(g(x))$. The inner function $g$ acts first. Composition models processes that happen in stages, such as converting currency and then applying a tax, and it is the structure the chain rule in calculus differentiates.

   • Evaluate from the inside out: compute $g(x)$, then apply $f$.
   • To build a formula, replace every $x$ in $f$ with the whole expression $g(x)$, in parentheses.
   • Decomposing a function, such as writing $\sqrt{x^2 + 1}$ as $f(g(x))$, is the reverse skill.
   Quick check: With $f(x) = 3x$ and $g(x) = x + 4$, find $f(g(2))$.

   $18$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study composition of functions

3. 3.Inverse Functions

   An inverse function reverses another: if $f$ sends 3 to 10, then $f^{-1}$ sends 10 back to 3. Composing them in either order returns the input, $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Graphically, the inverse is the reflection of the original graph across the line $y = x$.

   • To find $f^{-1}$: write $y = f(x)$, swap $x$ and $y$, solve for $y$.
   • The domain of $f^{-1}$ is the range of $f$, and vice versa.
   • $f^{-1}(x)$ is not $\frac{1}{f(x)}$.
   Quick check: If $f(x) = 4x + 7$, find $f^{-1}(15)$.

   $2$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study inverse functions

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Sequences and Series

Arithmetic and geometric sequences and sums.

• Precalculus overview
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