Premed · Premed · Calculus 1
Lecture 1: Functions and Their Graphs
Calculus I — Differential Calculus
Learning Objectives
By the end of this lecture, students will be able to:
- Define a function and identify its domain and range
- Use the vertical line test to determine whether a relation is a function
- Recognize and sketch the graphs of common families of functions
- Apply transformations (shifts, stretches, reflections) to basic graphs
- Compose functions and determine the domain of a composite function
- Classify functions as even, odd, or neither using symmetry
Lecture Content
I. What Is a Function?
A function f is a rule that assigns to each element x in a set D (the domain) exactly one element f(x) in a set called the range. We write f : D -> R, or simply y = f(x), where x is the independent variable (the input) and y = f(x) is the dependent variable (the output).
The domain is the set of all permissible inputs. Unless otherwise stated, the domain is taken to be the largest set of real numbers for which the formula makes sense. When determining the domain, watch for three common restrictions: division by zero, square roots of negative numbers, and logarithms of non-positive values.
The range is the set of all output values that the function actually attains. To illustrate domain restrictions, consider the following examples. The function f(x) = 1/(x - 2) has a domain of all real numbers except x = 2, because substituting x = 2 would produce division by zero. The function g(x) = sqrt(x + 3) requires x + 3 >= 0, so its domain is x >= -3. The function h(x) = ln(x) is only defined for positive inputs, giving it a domain of x > 0.
II. Representations of Functions
Functions can be represented in four fundamental ways. The algebraic representation uses a formula, such as y = x^2 + 1. The numerical representation presents a table of (x, y) pairs. The graphical representation plots the function in the Cartesian plane. The verbal representation describes the relationship in words, such as "the area of a circle as a function of its radius."
The graph of f is the set of all points (x, f(x)) where x lies in the domain. A natural question arises: given a curve in the xy-plane, how can we tell whether it represents a function? The answer is the Vertical Line Test, which states that a curve is the graph of a function if and only if no vertical line intersects the curve more than once. A parabola passes this test, while a circle does not, since a vertical line can cross a circle at two points.
<image>Panel A: A graph passing the vertical line test — a parabola y = x^2 with a vertical dashed line crossing it at exactly one point. Panel B: A circle x^2 + y^2 = 1 failing the vertical line test, with a vertical dashed line crossing it at two points. Both panels labeled clearly with axes and annotations.</image>
III. Catalog of Essential Functions
Several families of functions appear repeatedly throughout calculus. Linear functions take the form f(x) = mx + b, where m is the slope and b is the y-intercept; their graphs are straight lines. Power functions have the form f(x) = x^n, and different values of n produce familiar shapes: n = 1 gives a line, n = 2 gives a parabola, n = 3 gives a cubic curve, n = 1/2 gives the square root function, and n = -1 gives the reciprocal function.
Polynomial functions are expressions of the form f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0. The degree n determines the end behavior of the graph, and all polynomials produce smooth, continuous curves. Rational functions are ratios of polynomials, f(x) = P(x)/Q(x). They exhibit vertical asymptotes where Q(x) = 0 (provided P(x) is nonzero there), and their horizontal asymptotes are determined by comparing the degrees of P and Q.
The remaining essential families include trigonometric functions (sin x, cos x, tan x, and their reciprocals), exponential functions of the form f(x) = a^x (where a > 0 and a is not 1), and logarithmic functions f(x) = log_a(x), which serve as inverses of exponentials.
<image>A 2x3 grid of small plots showing the six essential function families: (1) linear y = 2x + 1, (2) quadratic y = x^2, (3) cubic y = x^3, (4) square root y = sqrt(x), (5) exponential y = 2^x, (6) logarithmic y = log_2(x). Each plot has labeled axes, the function formula, and key features (intercepts, asymptotes) marked.</image>
IV. Transformations of Functions
Starting from a known graph y = f(x), we can build new graphs through a set of standard transformations. Vertical shifts move the graph up or down: y = f(x) + c shifts the graph up by c units, while y = f(x) - c shifts it down by c units. Horizontal shifts move the graph left or right: y = f(x - c) shifts the graph right by c units, and y = f(x + c) shifts it left by c units.
Vertical stretching and compression are controlled by a multiplicative constant: y = c * f(x) stretches the graph vertically by a factor of c when c > 1, and compresses it when 0 < c < 1. Horizontal stretching and compression work through the argument: y = f(cx) compresses the graph horizontally by a factor of c when c > 1, and stretches it when 0 < c < 1.
Reflections flip the graph across an axis: y = -f(x) reflects the graph about the x-axis, while y = f(-x) reflects it about the y-axis. When combining multiple transformations, the order in which they are applied matters and must be tracked carefully.
<image>A demonstration of transformations applied to f(x) = x^2. Four panels: Panel A shows the base parabola y = x^2. Panel B shows y = (x - 2)^2 + 3 (shifted right 2, up 3) with arrows indicating the shift directions. Panel C shows y = -x^2 (reflection over x-axis). Panel D shows y = 2x^2 and y = (1/2)x^2 overlaid on the same axes showing vertical stretch and compression, with the original parabola in dashed gray for reference.</image>
V. Composition of Functions
Given two functions f and g, the composite function f composed with g is defined by (f o g)(x) = f(g(x)). This is read "f of g of x," and the key idea is that g is applied first, followed by f. The domain of f o g consists of all x in the domain of g for which g(x) lies in the domain of f. For example, if f(x) = sqrt(x) and g(x) = x - 3, then (f o g)(x) = sqrt(x - 3), and the domain is x >= 3, since we need x - 3 >= 0.
An important property is that composition is not commutative in general: f o g and g o f are typically different functions. Composition also works in reverse as a tool for understanding complicated functions. For instance, the function h(x) = (2x + 1)^5 can be decomposed as f(g(x)) where g(x) = 2x + 1 and f(u) = u^5. This decomposition skill becomes essential when applying the chain rule later in the course.
VI. Symmetry: Even and Odd Functions
A function is even if f(-x) = f(x) for all x in its domain, which means its graph is symmetric about the y-axis. Common examples include x^2, cos x, and |x|. A function is odd if f(-x) = -f(x) for all x in its domain, which means its graph is symmetric about the origin. Common examples include x^3, sin x, and tan x. Most functions are neither even nor odd. Recognizing symmetry can simplify graphing and computation significantly, especially when integrating over symmetric intervals.
VII. Piecewise-Defined Functions
Piecewise-defined functions use different formulas on different intervals. For example, f(x) = { x^2 if x < 0; x + 1 if x >= 0 } uses a quadratic rule for negative inputs and a linear rule for non-negative inputs. The absolute value function |x| = { x if x >= 0; -x if x < 0 } is a particularly important piecewise function that appears throughout calculus.
When graphing piecewise functions, it is important to check carefully whether each endpoint is included (indicated by a closed dot) or excluded (indicated by an open dot). A piecewise function can still be continuous provided the pieces agree at the boundary points where the formula changes.
