Functions, Their Properties, and Inverses: Calculus Foundations

Functions and Their Properties

Domain and Range

Understanding the domain and range of a function is fundamental in calculus. The domain is the set of all possible input values (x-coordinates) for which the function is defined, while the range is the set of all possible output values (y-coordinates) the function can produce.

• Domain of f(x): Set of all x you can plug in.

• Range of f(x): Set of all possible values of f(x).

• Example: For $f(x) = x^2 + 4$:

  • Domain: $(-\infty, \infty)$

  • Range: $[4, \infty)$

• Example: For $l(x) = \sqrt{x+4}$:

  • Domain: $[-4, \infty)$

  • Range: $[0, \infty)$

Graphical Representation: The graph of $y = \sqrt{x}$ starts at the origin and increases, while $y = \sqrt{x+4}$ is shifted left by 4 units.

Function Operations

Functions can be combined using addition, subtraction, multiplication, division, and composition.

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

• Example: If $f(x) = \sqrt{x+4}$ and $g(x) = x^2 - 3$, then:

  • $(f \circ g)(x) = f(g(x)) = \sqrt{(x^2 - 3) + 4} = \sqrt{x^2 + 1}$

  • $(g \circ f)(x) = g(f(x)) = (\sqrt{x+4})^2 - 3 = x + 4 - 3 = x + 1$

• Note: Composition is not generally commutative: $f \circ g \neq g \circ f$.

Symmetry of Functions

Functions can be classified as even or odd based on their symmetry properties.

• Even Function: $f(-x) = f(x)$ for all $x$. The graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$. The graph is symmetric about the origin.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

• Sum of Functions:

  • Sum of two even functions is even.

  • Sum of two odd functions is odd.

Difference Quotient

The difference quotient is a fundamental concept for understanding the derivative, representing the average rate of change of a function over an interval.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Interpretation: The slope of the secant line between $x$ and $x+h$ on the graph of $f(x)$.

• Example: For $f(x) = 2x^2$:

  • $f(x+h) = 2(x+h)^2 = 2x^2 + 4xh + 2h^2$

  • Difference quotient: $\frac{2x^2 + 4xh + 2h^2 - 2x^2}{h} = \frac{4xh + 2h^2}{h} = 4x + 2h$

Types and Graphs of Functions

Basic Function Types

Several fundamental functions are commonly encountered in calculus. Their graphs and properties are essential for understanding more advanced topics.

• Linear: $f(x) = x$

• Quadratic: $f(x) = x^2$

• Cubic: $f(x) = x^3$

• Absolute Value: $f(x) = |x|$

• Reciprocal: $f(x) = \frac{1}{x}$

• Exponential: $f(x) = e^x$

• Logarithmic: $f(x) = \ln x$

• Square Root: $f(x) = \sqrt{x}$

• Sine: $f(x) = \sin x$

• Cosine: $f(x) = \cos x$

• Tangent: $f(x) = \tan x$

Graphical Comparison: These functions have distinct shapes and symmetries. For example, quadratics are parabolas, cubics have inflection points, and exponentials grow rapidly.

Power Functions

A power function is of the form $f(x) = x^n$, where $n$ is a real number.

• Even powers: $y = x^n$ (n even, positive) are symmetric about the y-axis.

• Odd powers: $y = x^n$ (n odd, positive) are symmetric about the origin.

• Fractional powers: $y = x^{1/2} = \sqrt{x}$

• Examples:

  • $y = x^4$ (even)

  • $y = x^3$ (odd)

  • $y = |x|$ (absolute value)

  • $y = \frac{1}{x}$ (reciprocal, odd)

  • $y = \frac{1}{x^2}$ (reciprocal squared, even)

Exponential and Logarithmic Functions

Exponential Functions

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

• If $b > 1$, the function increases rapidly.

• If $0 < b < 1$, the function decreases rapidly.

• Example: $f(x) = (\frac{1}{2})^x$ is a decreasing exponential.

• Natural Exponential Function: $f(x) = e^x$

Inverse Functions

The inverse of a function $f(x)$, denoted $f^{-1}(x)$, "reverses" the effect of $f(x)$. A function has an inverse if it is one-to-one (injective).

• One-to-one test: $f(a) \neq f(b)$ whenever $a \neq b$.

• Domain of $f^{-1}(x)$: Range of $f(x)$.

• Range of $f^{-1}(x)$: Domain of $f(x)$.

• Graphical property: The graph of $f^{-1}(x)$ is the reflection of $f(x)$ about the line $y = x$.

• Example: If $f(x) = x^2 + 3$, $x \geq 0$, then $f^{-1}(x) = \sqrt{x - 3}$, with domain $[3, \infty)$ and range $[0, \infty)$.

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

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The general form is $f(x) = \log_b x$, where $b > 0$, $b \neq 1$.

• Definition: $\log_b x = y$ means $b^y = x$.

• Examples:

  • $\log_2 16 = 4$ because $2^4 = 16$

  • $\log_2 (\sqrt{2}) = \frac{1}{2}$ because $2^{1/2} = \sqrt{2}$

  • $\log_2 (\frac{1}{8}) = -3$ because $2^{-3} = \frac{1}{8}$

• Properties of Logarithms:

  • $\log_b 1 = 0$

  • $\log_b b = 1$

  • $\log_b (b^x) = x$

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b (\frac{x}{y}) = \log_b x - \log_b y$

  • $\log_b (x^n) = n \log_b x$

Summary Table: Common Function Types and Their Graphs

Function Type	Equation	Graph Shape	Symmetry
Linear	$f(x) = x$	Straight line	Odd
Quadratic	$f(x) = x^2$	Parabola	Even
Cubic	$f(x) = x^3$	S-shaped curve	Odd
Absolute Value	$f(x) = |x|$	V-shape	Even
Reciprocal	$f(x) = \frac{1}{x}$	Hyperbola	Odd
Exponential	$f(x) = e^x$	Rapidly increasing curve	Neither
Logarithmic	$f(x) = \ln x$	Slowly increasing curve	Neither
Square Root	$f(x) = \sqrt{x}$	Starts at origin, increases	Neither
Sine	$f(x) = \sin x$	Wave	Odd
Cosine	$f(x) = \cos x$	Wave	Even
Tangent	$f(x) = \tan x$	Repeating curve with asymptotes	Odd

Additional info: These notes cover foundational concepts from Chapter 1 (Functions) and introduce ideas essential for later calculus topics, such as limits and derivatives.