Foundations of Functions and Sequences in Calculus I

Functions: Definitions and Properties

Definition and Notation

A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the codomain. Functions are fundamental objects in calculus, used to model relationships between varying quantities.

• Input Form: $y = f(x)$

• Explicit Form: $y = 2x + 1$

• Implicit Form: $x^2 + y^2 = 1$

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Domain and Range

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

• Range: The set of all possible output values ($f(x)$) the function can produce.

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

Types of Functions

• Algebraic Functions: Built from algebraic operations (addition, subtraction, multiplication, division, roots).

• Piecewise-Defined Functions: Defined by different expressions over different intervals.

• Even and Odd Functions:

  • Even: $f(-x) = f(x)$ (symmetric about the y-axis)

  • Odd: $f(-x) = -f(x)$ (symmetric about the origin)

Identifying the Graphs of Functions

• Use the vertical line test: a graph represents a function if any vertical line crosses it at most once.

• Common function graphs:

  • Linear: $f(x) = x$

  • Quadratic: $f(x) = x^2$

  • Cubic: $f(x) = x^3$

  • Absolute Value: $f(x) = |x|$

  • Square Root: $f(x) = \sqrt{x}$

  • Reciprocal: $f(x) = \frac{1}{x}$

Operations on Functions

• Sum: $(f+g)(x) = f(x) + g(x)$

• Difference: $(f-g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

• Composition: $(f \circ g)(x) = f(g(x))$

Transformations of Functions

• Vertical Shifts: $f(x) + c$ shifts up by $c$ units.

• Horizontal Shifts: $f(x + c)$ shifts left by $c$ units.

• Reflections: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Stretching/Compressing: $af(x)$ stretches vertically by $a$; $f(ax)$ compresses horizontally by $a$.

Inverse Functions

Definition and Properties

An inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. A function has an inverse if and only if it is one-to-one (passes the horizontal line test).

• Definition: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

• Finding the Inverse: Swap $x$ and $y$ in $y = f(x)$, then solve for $y$.

• Example: $f(x) = 2x + 3 \implies f^{-1}(x) = \frac{x-3}{2}$

Domain and Range of Inverse Functions

• The domain of $f$ becomes the range of $f^{-1}$, and vice versa.

Exponential and Logarithmic Functions

Exponential Functions

An exponential function has the form $f(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Properties: Always positive, increasing if $a > 1$, decreasing if $0 < a < 1$.

• Example: $f(x) = 2^x$

Logarithmic Functions

The logarithmic function is the inverse of the exponential function: $y = \log_a x$ if and only if $a^y = x$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Properties: $\log_a 1 = 0$, $\log_a a = 1$

• Change of Base Formula: $\log_a x = \frac{\log_b x}{\log_b a}$

• Example: $\log_{10} 100 = 2$

Sequences and Summation

Sequences

A sequence is an ordered list of numbers, often defined by a formula for the $n$th term $a_n$.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Example: $2, 4, 6, 8, ...$ is arithmetic with $d=2$

Summation Notation

• Definition: $\sum_{k=1}^n a_k$ denotes the sum $a_1 + a_2 + ... + a_n$

• Properties:

  • $\sum_{k=1}^n c = nc$

  • $\sum_{k=1}^n k = \frac{n(n+1)}{2}$

  • $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$

Finding Sums of Sequences

• Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

• Example: $\sum_{k=1}^{10} k = 55$

Table: Common Functions and Their Properties

Function	Equation	Domain	Range	Even/Odd
Linear	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$	Odd
Quadratic	$f(x) = x^2$	$(-\infty, \infty)$	$(0, \infty)$	Even
Cubic	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$	Odd
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$(0, \infty)$	Even
Square Root	$f(x) = \sqrt{x}$	$(0, \infty)$	$(0, \infty)$	Neither
Reciprocal	$f(x) = \frac{1}{x}$	$x \neq 0$	$x \neq 0$	Odd

Additional info:

• Some content, such as the Binomial Theorem, is referenced but not fully detailed in the notes. For completeness, the Binomial Theorem states:

  • $(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

• All examples and properties are standard for introductory calculus courses.