Functions and Change: Foundations of Calculus (Section 1.1 Study Notes)

Functions and Change

Section 1.1: What is a Function

Understanding functions is fundamental in calculus, as they describe how one quantity depends on another. This section introduces the concept of a function, its notation, properties, and graphical representation.

Definition of a Function

• Function: A function is a rule that assigns exactly one output to each input from a given set.

• Formal Definition: A function f from a set A to a set B assigns to every input number x in A exactly one corresponding number f(x) in B as output.

• Domain: The set of all possible input values (x) for which the function is defined.

• Range: The set of all output values (f(x)) that result from applying the function to the domain.

• Function Notation: y = f(x) means "y is a function of x"; x is the independent variable, y is the dependent variable.

Key Properties of Functions

• Each input has one and only one output.

• Different inputs may have the same output.

• An input cannot have more than one output.

Examples of Functions

• Linear Function: $y = f(x) = 3x + 6$

• Quadratic Function: $y = f(x) = x^2 + 1$

• Square Root Function: $y = f(x) = \sqrt{x}$

• Rational Function: $y = f(t) = \frac{1}{t-1}$

Domain and Range

• Domain: The set of all possible input values for which the function is defined. Graphically, this is the part of the horizontal x-axis covered by the function.

• Range: The set of all possible output values. Graphically, this is the part of the vertical y-axis covered by the function.

Example: For $y = f(x) = 3x + 6$, the domain and range are both all real numbers ($\mathbb{R}$).

Example: For $y = f(x) = \sqrt{x}$, the domain is $x \geq 0$ and the range is $y \geq 0$.

Evaluating Functions

• Meaning of $f(a)$: Substitute $x = a$ into the function.

• Example: If $y = f(x) = 2x + 4$, then $f(0) = 4$, $f(1) = 6$, $f(3) = 10$, $f(5) = 14$, $f(-4) = -4$.

Graphical Representation and the Vertical Line Test

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Example: The graph of $y = x^2$ passes the vertical line test, but the graph of $x^2 + y^2 = 1$ (a circle) does not.

Ways to Represent a Function (The Rule of Four)

• Tables (numerically)

• Graphs (visually)

• Formulas (algebraically)

• Words (verbally)

Intercepts

• y-intercept: The value of $y$ where the graph crosses the y-axis. To find, set $x = 0$ and solve for $y$.

• x-intercept: The value of $x$ where the graph crosses the x-axis. To find, set $y = 0$ and solve for $x$.

• Example: For $y = f(x) = x^2 + 2$, the y-intercept is $2$(when$x = 0$), and the x-intercepts are solutions to$x^2 + 2 = 0$ (no real solutions).

Increasing and Decreasing Functions

• Increasing Function: $f$ is increasing if $f(x_2) > f(x_1)$ whenever $x_2 > x_1$.

• Decreasing Function: $f$ is decreasing if $f(x_2) < f(x_1)$ whenever $x_2 > x_1$.

Applications and Examples

• Distance as a Function of Time: If a motorist travels at 70 km/h for 2 hours and then at 80 km/h for $t$ hours, the distance $d$ travelled can be expressed as a function of $t$.

• Population as a Function of Year: If $P = f(t)$ gives the population in millions at year $t$, then $f(2005) = 12$ means the population in 2005 was 12 million.

• Bank Account Example: The balance $B$ in an account as a function of time $t$ can be analyzed to find the original deposit, estimate values at specific times, and determine when a certain balance is reached.

Tables: Function Representation and Evaluation

x	0	1	2	3	4
f(x)	1	4	7	15	18

Purpose: This table numerically represents a function by listing input-output pairs. To determine if it is a function, check that each input $x$ has only one output $f(x)$.

Additional info: These foundational concepts are essential for understanding more advanced calculus topics such as limits, derivatives, and integrals.