M151Q Precalculus · Unit 1: Functions, graphs, models · on Exam 1
MFG 2.1–2.2 Nonlinear models and the basic functions
Study notes for Nonlinear models and the basic functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Many quantities are not linear. Build a model from a geometric or verbal relationship, then use the graph or algebra: area of a square from its perimeter $A = (P/4)^2$, volume of a box from its side, revenue from price.
- Solve nonlinear equations by extraction of roots ($x^2 = 20 \Rightarrow x = \pm\sqrt{20}$), by factoring, or graphically (intersection of $y = f(x)$ with $y = c$). Keep both roots unless the context excludes one.
- Library of basic functions you must know by shape, domain and range: $y = x^2$, $x^3$, $\sqrt{x}$, $\sqrt[3]{x}$, $1/x$, $1/x^2$, $|x|$. Know which are even/odd, increasing/decreasing, and their asymptotes.
- Piecewise functions use different formulas on different intervals; evaluate with the piece whose condition holds. A jump where the pieces do not meet means the graph is discontinuous there.
Key formulas
Basic functions
$$x^2\ [0,\infty);\quad x^3\ \mathbb R;\quad \sqrt x\ [0,\infty)\to[0,\infty);\quad \sqrt[3]{x}\ \mathbb R;\quad \tfrac1x\ x\ne0;\quad \tfrac1{x^2}\ y>0;\quad |x|\ y\ge0$$
Extraction of roots
$$x^2 = k\ (k>0) \Rightarrow x = \pm\sqrt k$$
Worked example
Let $f(x) = \begin{cases} x + 2 & x < 1 \\ 4 - x & x \ge 1\end{cases}$. Find $f(0)$, $f(1)$, $f(3)$ and decide whether the graph has a break at $x = 1$.
Show the solution
$f(0) = 2$ (first piece), $f(1) = 3$ (second piece, since $1 \ge 1$), $f(3) = 1$. Near $x=1$ the first piece approaches $3$ and the second equals $3$: the pieces meet, no break.
Common mistakes
- Dropping the negative root when solving x² = k with no context.
- Using the wrong piece at the boundary of a piecewise function (check ≤ vs <).
- Confusing 1/x (odd, two branches) with 1/x² (even, both branches above the axis).
Exam tipSketch the seven basic functions from memory once a week until it is automatic. Every transformation question starts from one of them.
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