The Timpson Lyceum · Calculus 1 (2024)

Review Functions, Function Operations, & Graphing - Lecture 2 (Calculus 1)

Overview

This lecture reviews the foundational language of functions that the rest of calculus is built on, working through sections 1.1 and 1.2 of the text. It defines what a function is, how to read off its domain and range, and how to recognize and graph the standard families of functions you will use all year: linear, power, polynomial, rational, exponential, logarithmic, and trigonometric. It also covers the tools for describing and building functions — piecewise definitions, even/odd symmetry, increasing/decreasing behavior, the algebraic combinations (sum, difference, product, quotient, composition), and the geometric transformations (shifts, stretches, compressions, reflections). The material is mostly review, so the goal is to surface any gaps and fix them before calculus proper begins.

Key Concepts

Function, domain, and range

• A function $f$ from a set $D$ to a set $Y$ is a rule that assigns to each element $x \in D$ exactly one element $f(x) \in Y$.
• The domain $D$ is the set of allowed inputs; you must assign an output to every element of $D$.
• The range is the set of outputs actually produced — the subset of $Y$ that gets used. You need not use all of $Y$.
• In $y = f(x)$, $x$ is the independent variable and $y$ is the dependent variable ($y$'s value depends on $x$).

Natural domain

• When a domain is not stated, the natural domain is assumed: all real $x$ for which $f(x)$ is a real number.
• Only two things break a real-valued function: dividing by zero and taking the square root (an even root) of a negative number.
• Example results: $\sqrt{x}$ has domain $[0,\infty)$ and range $[0,\infty)$; $1/x$ has domain and range $(-\infty,0)\cup(0,\infty)$; $\sqrt{4-x}$ has domain $(-\infty,4]$.

Graph and the vertical line test

• The graph of $f$ is the set of points $(x, f(x))$ for all $x$ in the domain, plotted in the Cartesian ($xy$) plane.
• Vertical line test: a curve is the graph of a function only if every vertical line meets it at most once. (For example, the circle $x^2 + y^2 = 1$ fails — at $x=0$ both $y = 1$ and $y = -1$ occur.)

Piecewise functions

• A piecewise function applies different formulas on different parts of the domain.
• The absolute value is the classic example: $|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$
• Floor $\lfloor x \rfloor$ rounds down to the nearest integer; ceiling $\lceil x \rceil$ rounds up. E.g. $\lfloor 2.9 \rfloor = 2$, $\lfloor -1.7 \rfloor = -2$, $\lceil 1.4 \rceil = 2$.

Increasing / decreasing

• For $x_1 < x_2$ in an interval $I$: $f$ is increasing if $f(x_1) < f(x_2)$, and decreasing if $f(x_1) > f(x_2)$ — for every such pair in $I$.

Even / odd symmetry

• $f$ is even if $f(-x) = f(x)$ for all $x$ — symmetric about the $y$-axis (e.g. $x^2$, $x^2+1$).
• $f$ is odd if $f(-x) = -f(x)$ for all $x$ — symmetric about the origin (e.g. $x$, $x^3$).
• Most functions are neither (e.g. $x+1$). The constant $f(x)=0$ is the rare function that is both.

Equations of a line (all equivalent; choose by what you know)

• Two-point form: $y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
• Point-slope form: $y - y_1 = m(x - x_1)$
• Slope-intercept form: $y = mx + b$, where $b$ is the $y$-intercept (the $y$-value when $x = 0$).

Families of functions

• Linear: $f(x) = mx + b$. Special cases: the identity $f(x) = x$ (slope $1$, intercept $0$) and the constant $f(x) = c$ (slope $0$).
• Proportionality: $y$ is directly proportional to $x$ if $y = kx$ ($k \ne 0$); inversely proportional if $y = k/x$.
• Power functions: $f(x) = x^a$ for fixed $a$. Note $x^0 = 1$, $x^1 = x$, $x^2$ is a parabola, $x^3$ is odd-shaped; even powers are even functions, odd powers are odd functions.
• Polynomials: $f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$ — a sum of terms whose exponents are all whole numbers $\ge 0$.
• Rational functions: $f(x) = \dfrac{p(x)}{q(x)}$, a polynomial over a polynomial. (E.g. $\sqrt{x}/x$ is not rational, since $\sqrt{x} = x^{1/2}$.)
• Exponentials: $f(x) = a^x$ with $a > 0$, $a \ne 1$ — the variable is in the exponent. These grow far faster than power functions (compare $x^2$ vs $2^x$ at $x = 10$: $100$ vs $1024$).
• Logarithms: $\log_a c = b \iff a^b = c$. Logs undo exponentials: $\log_a(a^x) = x$ and $a^{\log_a x} = x$. They grow very slowly but are unbounded above; domain $(0,\infty)$, range all reals.

Sine and cosine graphs

• $\cos x$ is the $x$-coordinate and $\sin x$ the $y$-coordinate of the point at angle $x$ on the unit circle, so both stay in $[-1, 1]$.
• Both are periodic waves, repeating every $2\pi$.
• They differ by a shift: $\sin\!\left(x + \tfrac{\pi}{2}\right) = \cos x$ and $\cos\!\left(x - \tfrac{\pi}{2}\right) = \sin x$.

Combining functions

• Algebraic: $(f+g)(x) = f(x)+g(x)$, and likewise $f-g$, $fg$, $f/g$. The domain is the intersection of the two domains (and for $f/g$, also exclude points where $g(x) = 0$).
• Composition: $(f \circ g)(x) = f(g(x))$ — apply $g$ first, then $f$. In general $f \circ g \ne g \circ f$. The domain is the inputs in the domain of $g$ whose outputs land in the domain of $f$.

Transformations (for constant $k, c > 0$)

• $f(x) + k$ shifts up $k$; $f(x) - k$ shifts down $k$.
• $f(x - k)$ shifts right $k$; $f(x + k)$ shifts left $k$.
• $c\,f(x)$ is a vertical stretch ($c>1$) or compression ($0<c<1$).
• $f(cx)$ is a horizontal compression ($c>1$) or stretch ($0<c<1$).
• $-f(x)$ reflects across the $x$-axis; $f(-x)$ reflects across the $y$-axis.

Worked Example

Composition and its domain. Let $f(x) = \sqrt{x}$ and $g(x) = x + 1$. Compute $(f \circ g)(x)$ and find its domain. The rule is $(f \circ g)(x) = f(g(x))$, evaluate $g$ first, then substitute into $f$:

The domain is not all reals: even though $g$ accepts every real number, only inputs with $x \ge -1$ produce an output $g(x) \ge 0$ that $f = \sqrt{\,\cdot\,}$ can accept.

Summary

• A function assigns each domain input exactly one output; the range is the set of outputs actually used, and the natural domain is every input that keeps the output real (avoid dividing by zero and even roots of negatives).
• The vertical line test distinguishes function graphs from non-functions; piecewise definitions (absolute value, floor, ceiling) handle functions built from several formulas.
• Classify symmetry with even ($f(-x)=f(x)$) and odd ($f(-x)=-f(x)$), and monotonicity with increasing/decreasing comparisons on an interval.
• Lines come in two-point, point-slope, and slope-intercept forms; the major function families are linear, power, polynomial, rational, exponential, logarithmic, and trigonometric — with exponentials growing fastest and logarithms slowest.
• $\sin x$ and $\cos x$ are unit-circle waves bounded in $[-1,1]$, period $2\pi$, and offset from each other by $\tfrac{\pi}{2}$.
• Functions combine by $+,-,\times,\div$ (intersect the domains) and by composition $f\circ g$ (order matters, and the domain must respect both inner and outer functions); transformations shift, stretch, compress, and reflect a known graph.