Real Functions and Their Graphs

Real Functions: Definition

While the concept of a function can apply to any sets (mapping students to their roll numbers, countries to their capitals, etc.), a particularly important class of functions in mathematics involves sets of real numbers. These are known as real functions, and they form the basis of calculus and mathematical analysis.

Real-Valued Function

A function $f: A \to B$ is called a real-valued function if its codomain $B$ is a subset of the set of real numbers $\mathbb{R}$ ($B \subseteq \mathbb{R}$).

This means that the output $f(x)$ of the function is always a real number. The domain $A$, however, can be any set (e.g., a set of complex numbers, vectors, functions, etc., depending on the context). For most common applications in introductory mathematics, the domain is also a subset of $\mathbb{R}$.

Example:

Let $A = \{\text{Apple, Banana, Cherry}\}$. Define a function $w: A \to \mathbb{R}_{\ge 0}$ that maps each fruit to its weight in grams. If $w(\text{Apple})=150$ g, $w(\text{Banana})=120$ g, $w(\text{Cherry})=5$ g, this is a real-valued function because the codomain (non-negative real numbers, a subset of $\mathbb{R}$) contains real numbers as outputs, even though the domain is a set of fruits.

Real Function (or Real Function of a Real Variable)

A function $f: A \to B$ is called a real function (or a real function of a real variable) if both its domain $A$ and its codomain $B$ are subsets of the set of real numbers $\mathbb{R}$.

These are the functions that are typically encountered when studying graphs, limits, derivatives, and integrals. The input is a real number, and the output is a real number.

Examples:

• $f(x) = 2x + 3$. Here, the domain could be $\mathbb{R}$, and the codomain could be $\mathbb{R}$.
• $g(x) = x^2$. Domain $\subseteq \mathbb{R}$, Codomain $\subseteq \mathbb{R}$.
• $h(x) = \frac{1}{x}$. Domain is usually $\mathbb{R} \setminus \{0\}$, Codomain $\subseteq \mathbb{R}$.
• $k(x) = \sqrt{x}$. Domain is usually $[0, \infty)$, Codomain $\subseteq \mathbb{R}$.

When a real function is defined by a formula (e.g., $f(x) = \frac{1}{x-1}$), and the domain and codomain are not explicitly specified, it is a common convention to assume the domain is the largest possible subset of $\mathbb{R}$ for which the formula produces a real number output, and the codomain is the set of all real numbers, $\mathbb{R}$. This largest possible domain is sometimes called the "natural domain" or "domain of definition".

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Domain and Range of a Real Function (Techniques for Finding)

For a real function $f(x)$, determining its domain and range is crucial for understanding where the function is defined and what values it can possibly produce. The domain is the set of valid inputs, and the range is the set of actual outputs.

Finding the Domain of a Real Function

The domain of a real function $f(x)$ is the set of all real numbers $x$ for which the expression $f(x)$ is defined and yields a real number output. When given a formula for $f(x)$ without an explicitly stated domain, we find the "natural domain".

Steps and Common Restrictions:

1. Assume the domain is all real numbers ($\mathbb{R}$) unless there are specific restrictions based on the function's formula.
2. Identify potential sources of restrictions that would result in an undefined or non-real output:
   • Division by Zero: If the function involves a fraction $\frac{g(x)}{h(x)}$, the denominator $h(x)$ cannot be zero. Exclude any $x$ values for which $h(x)=0$. Solve $h(x) = 0$ and remove the solutions from $\mathbb{R}$.
   • Even Roots of Negative Numbers: If the function involves an even root like a square root ($\sqrt{g(x)}$), fourth root ($\sqrt[4]{g(x)}$), etc., the expression inside the root, $g(x)$, must be non-negative. Solve the inequality $g(x) \ge 0$ for $x$. The domain must be limited to the values satisfying this inequality.
   • Logarithms of Non-positive Numbers: If the function involves a logarithm like $\log_a(g(x))$ (where $a > 0, a \neq 1$), the argument $g(x)$ must be strictly positive. Solve the inequality $g(x) > 0$ for $x$.
   • Trigonometric Functions: Functions like $\tan(x)$ are undefined where $\cos(x)=0$, i.e., $x = \frac{\pi}{2} + n\pi$ for integer $n$. $\cot(x)$ is undefined where $\sin(x)=0$, i.e., $x = n\pi$ for integer $n$. $\text{cosec}(x)$ requires $\sin(x) \neq 0$, $\sec(x)$ requires $\cos(x) \neq 0$.
3. If there are multiple restrictions, the domain is the set of all real numbers that satisfy ALL restrictions simultaneously. This means finding the intersection of the domains required by each individual restriction.

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Finding the Range of a Real Function

The range of a real function $f(x)$ is the set of all possible real output values $y = f(x)$ that the function can produce when $x$ takes on all values from its domain. Finding the range can be more involved than finding the domain.

Techniques for Finding the Range:

1. Algebraic Method (Solving for x):

   • Set $y$ equal to the function's rule: $y = f(x)$.
   • Try to solve this equation for $x$ in terms of $y$.
   • Analyze the expression for $x$ in terms of $y$. Determine for which values of $y$ a corresponding real value of $x$ exists within the domain of $f$. The set of these $y$ values is the range. Look for algebraic restrictions on $y$ (like division by zero, even roots of negative numbers).

2. Graphical Method (Projection on y-axis):

   • Sketch the graph of the function $y = f(x)$ over its domain.
   • The range is the set of all y-values covered by the graph. Imagine "projecting" the graph onto the y-axis. The interval(s) or set of points covered on the y-axis constitute the range.
   • Identify minimum and maximum y-values (if they exist) and whether the graph is continuous between them.

3. Using Properties of Standard Functions:

   • Leverage known properties of common functions (e.g., $x^2 \ge 0$, $|x| \ge 0$, $-1 \le \sin x \le 1$, $e^x > 0$, range of polynomial of odd degree is $\mathbb{R}$).
   • Understand how transformations (vertical/horizontal shifts, stretches, reflections) affect the range of a base function.

4. Calculus Methods:

   For complex functions, calculus (finding critical points, intervals of increase/decrease, limits) can be used to determine the minimum and maximum values and overall behavior, which helps identify the range.

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Graph of a Real Function

The graphical representation of a real function $f: A \to B$ (where $A, B \subseteq \mathbb{R}$) is a visual depiction of the relationship between the input values from the domain and their corresponding output values in the range.

Definition of the Graph of a Real Function

Let $f: A \to B$ be a real function with domain $A$. The graph of the function $f$ is defined as the set of all ordered pairs $(x, f(x))$ such that $x$ belongs to the domain of $f$.

Since both $x$ (the input from the domain $A \subseteq \mathbb{R}$) and $y = f(x)$ (the output, which is in the range, and thus in the codomain $B \subseteq \mathbb{R}$) are real numbers, each ordered pair $(x, y)$ in the graph can be plotted as a point in the Cartesian coordinate plane ($\mathbb{R}^2$). The collection of all such points forms the visual graph of the function.

Essentially, the graph is the set of all points $(x, y)$ on the Cartesian plane that satisfy the equation $y = f(x)$ for all valid input values $x$ from the function's domain.

The graph provides a geometric interpretation of the function's behavior, showing how the output varies as the input changes.

Graphical Representation of Functions (General)

The graphical representation transforms the abstract rule of a function into a visible shape on a coordinate system. This visualization is invaluable for understanding properties like increasing/decreasing intervals, maxima/minima, continuity, and identifying the domain and range visually.

Key Aspects of Graphical Representation:

• Coordinate System:

  A standard Cartesian coordinate plane is used, with the horizontal axis (x-axis) representing the values from the domain (input values) and the vertical axis (y-axis) representing the values from the codomain (output values).

• Plotting Points:

  Each ordered pair $(x, f(x))$ that belongs to the graph of the function corresponds to a unique point $P$ with coordinates $(x, y)$ in the Cartesian plane, where the x-coordinate is the input $x$ and the y-coordinate is the corresponding output $y = f(x)$.

• The Shape of the Graph:

  The collection of all such points $(x, y)$ for every $x$ in the domain forms a curve, a line, or sometimes a set of discrete points (if the domain is a discrete set like integers). This visual trace on the plane is what we call the "graph of the function".

• Interpretation:

  For any point $(x_0, y_0)$ that lies on the graph of a function $f$, it means that $y_0$ is the output of the function when the input is $x_0$, i.e., $y_0 = f(x_0)$.

• Vertical Line Test Revisited:

  A curve drawn in the Cartesian plane represents the graph of a function $y = f(x)$ if and only if every vertical line drawn on the plane intersects the curve at most once. This test visually confirms the fundamental property of a function: each input $x$ corresponds to a single, unique output $y$.

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Some Standard Real Functions (Identity, Modulus, etc.) and their Graphs

In the study of real functions, certain functions appear frequently and serve as building blocks for more complex functions. Understanding their definitions, domains, ranges, and especially their graphs is fundamental.

Here are some essential standard real functions:

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1. Identity Function

The identity function maps every real number to itself.

• Definition: $f(x) = x$
• Domain: $\mathbb{R}$ (any real number can be an input)
• Codomain: $\mathbb{R}$ (as a real function, the codomain is $\mathbb{R}$ unless specified otherwise)
• Range: $\mathbb{R}$ (every real number can be an output, $y=x$)
• Graph: This is the set of points $(x, x)$ for all $x \in \mathbb{R}$. The graph is a straight line passing through the origin $(0, 0)$ and making an angle of $45^\circ$ with the positive x-axis (since the slope is 1).
• Properties: The identity function is both injective ($x_1 = x_2 \implies f(x_1)=f(x_2)$) and surjective (Range=Codomain=$\mathbb{R}$), so it is bijective.

(Imagine a graph of the line $y=x$).

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2. Constant Function

A constant function maps every real number to the same fixed real number.

• Definition: $f(x) = c$, where $c$ is a fixed real number.
• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$
• Range: $\{c\}$ (The set containing only the value $c$. This is a singleton set).
• Graph: This is the set of points $(x, c)$ for all $x \in \mathbb{R}$. The graph is a horizontal line parallel to the x-axis, passing through the point $(0, c)$ on the y-axis.
• Properties: If the domain has more than one element, it is not injective (e.g., $f(1)=c, f(2)=c$, but $1 \neq 2$). It is not surjective unless the codomain is specifically defined as $\{c\}$.

(Imagine a graph of a horizontal line $y=c$).

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3. Modulus Function (Absolute Value Function)

The modulus function gives the absolute value of a real number.

• Definition: $f(x) = |x|$. It is defined piecewise as:

  $|x| = \begin{cases} \phantom{-}x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}$

• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$
• Range: $[0, \infty)$ (The set of all non-negative real numbers. The absolute value of any real number is always $\ge 0$).
• Graph: This is the set of points $(x, |x|)$ for all $x \in \mathbb{R}$. The graph consists of two rays originating from the origin $(0, 0)$: the line $y=x$ for $x \ge 0$ and the line $y=-x$ for $x < 0$. It forms a V-shape with its vertex at the origin.
• Properties: It is not injective (e.g., $|-2|=2, |2|=2$, but $-2 \neq 2$). It is not surjective onto $\mathbb{R}$ (since the range is only $[0, \infty)$).

(Imagine a graph of $y=|x|$).

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4. Signum Function

The signum function (or sign function) indicates the sign of a real number.

• Definition: $f(x) = \text{sgn}(x)$. It is defined piecewise as:

  $\text{sgn}(x) = \begin{cases} \phantom{-}1 & \text{if } x > 0 \\ \phantom{-}0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$

  It can also be defined as $f(x) = \frac{|x|}{x}$ for $x \neq 0$ and $f(0)=0$.

• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$
• Range: $\{-1, 0, 1\}$ (Only these three discrete values are possible outputs).
• Graph: Consists of three parts: a horizontal ray at $y=1$ for $x > 0$ (with an open circle at $(0, 1)$), a single point at $(0, 0)$, and a horizontal ray at $y=-1$ for $x < 0$ (with an open circle at $(0, -1)$). It's a step-like graph with jumps at $x=0$.
• Properties: It is not injective (many inputs map to 1, 0, or -1). It is not surjective onto $\mathbb{R}$.

(Imagine a graph of the signum function).

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5. Greatest Integer Function (Floor Function)

The greatest integer function (or floor function) gives the largest integer less than or equal to the input real number.

• Definition: $f(x) = [x]$ or $f(x) = \lfloor x \rfloor$. It gives the greatest integer $\le x$.
• Examples: $\lfloor 3.7 \rfloor = 3$, $\lfloor 4 \rfloor = 4$, $\lfloor -1.2 \rfloor = -2$, $\lfloor 0.9 \rfloor = 0$, $\lfloor -0.1 \rfloor = -1$.
• Domain: $\mathbb{R}$
• Codomain: $\mathbb{R}$
• Range: $\mathbb{Z}$ (The set of all integers. The output is always an integer).
• Graph: A series of horizontal line segments. For any integer $n$, $f(x) = n$ for all $x$ in the interval $[n, n+1)$. Each segment starts at an integer point on the left (closed circle) and extends up to, but not including, the next integer (open circle). It's a "step" function.
• Properties: It is not injective (e.g., $\lfloor 2.1 \rfloor = 2, \lfloor 2.5 \rfloor = 2$, but $2.1 \neq 2.5$). It is not surjective onto $\mathbb{R}$ (since the range is $\mathbb{Z}$).

(Imagine a graph of the greatest integer function).

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6. Polynomial Functions

Polynomial functions are built using only addition, subtraction, multiplication, and non-negative integer exponents of the variable.

• Definition: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer (the degree), and $a_0, a_1, \dots, a_n$ are real coefficients with $a_n \neq 0$ for $n > 0$. (If $n=0$, it's a constant function, $f(x)=a_0$).
• Examples:
  • Linear function ($n=1$): $f(x) = mx + c$ (Graph is a straight line).
  • Quadratic function ($n=2$): $f(x) = ax^2 + bx + c$ (Graph is a parabola).
  • Cubic function ($n=3$): $f(x) = ax^3 + bx^2 + cx + d$.
• Domain: $\mathbb{R}$ (Polynomials are defined for all real numbers).
• Codomain: $\mathbb{R}$
• Range: Depends on the degree $n$ and the leading coefficient $a_n$.
  • If the degree $n$ is odd, the graph extends infinitely upwards and downwards. The range is $\mathbb{R}$.
  • If the degree $n$ is even, the graph has a minimum or maximum value. The range is an interval of the form $[k, \infty)$ or $(-\infty, k]$ for some real number $k$.
• Graph: Polynomial graphs are smooth, continuous curves with no sharp corners, breaks, or vertical asymptotes. The shape is determined by the degree.

(Imagine graphs of $y=x$, $y=x^2$, $y=x^3$).

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7. Rational Functions

Rational functions are ratios of polynomial functions.

• Definition: $f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions, and $q(x)$ is not the zero polynomial.
• Domain: $\mathbb{R} \setminus \{x \mid q(x) = 0\}$ (All real numbers except the roots of the denominator polynomial, as division by zero is undefined).
• Codomain: $\mathbb{R}$
• Range: Varies significantly depending on the specific polynomials $p(x)$ and $q(x)$. Can be found using calculus (finding extrema) or analyzing horizontal/vertical asymptotes.
• Graph: Can have vertical asymptotes at the x-values where the denominator is zero (and the numerator is not zero). Can have horizontal or oblique asymptotes depending on the degrees of $p(x)$ and $q(x)$. The graph may have breaks at asymptotes.

(Imagine a graph of $y=1/x$).

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8. Exponential Function

Exponential functions have a constant base raised to a variable power.

• Definition: $f(x) = a^x$, where the base $a$ is a positive constant and $a \neq 1$. (If $a=1$, $f(x)=1^x=1$, which is a constant function).
• Domain: $\mathbb{R}$ (Any real number can be an exponent).
• Codomain: $\mathbb{R}$
• Range: $(0, \infty)$ (The set of all positive real numbers. The output of $a^x$ is always positive for $a>0$).
• Graph: Passes through the point $(0, a^0=1)$. If $a > 1$, the graph increases rapidly from left to right (exponential growth). If $0 < a < 1$, the graph decreases rapidly from left to right (exponential decay). The x-axis ($y=0$) is a horizontal asymptote. The graph is always above the x-axis.
• Properties: An exponential function $f: \mathbb{R} \to (0, \infty)$ is bijective.

(Imagine graphs of $y=2^x$ and $y=(1/2)^x$).

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9. Logarithmic Function

Logarithmic functions are the inverse functions of exponential functions.

• Definition: $f(x) = \log_a x$, where the base $a$ is a positive constant and $a \neq 1$. The definition $y = \log_a x$ is equivalent to $a^y = x$.
• Domain: $(0, \infty)$ (The input must be strictly positive, as the range of the exponential function is $(0, \infty)$).
• Codomain: $\mathbb{R}$
• Range: $\mathbb{R}$ (The range of $\log_a x$ is the domain of $a^x$).
• Graph: Passes through the point $(a, \log_a a = 1)$ and $(1, \log_a 1 = 0)$. If $a > 1$, the graph increases slowly from left to right. If $0 < a < 1$, the graph decreases from left to right. The y-axis ($x=0$) is a vertical asymptote. The graph is always to the right of the y-axis.
• Properties: A logarithmic function $f: (0, \infty) \to \mathbb{R}$ is bijective.

(Imagine graphs of $y=\log_2 x$ and $y=\log_{1/2} x$).

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10. Square Root Function

The square root function gives the principal (non-negative) square root of a non-negative number.

• Definition: $f(x) = \sqrt{x}$.
• Domain: $[0, \infty)$ (The input must be non-negative for the output to be a real number).
• Codomain: $\mathbb{R}$
• Range: $[0, \infty)$ (The principal square root is always non-negative. Any non-negative number $y$ can be achieved as $\sqrt{x}$ by setting $x=y^2$, and if $y \ge 0$, $x=y^2 \ge 0$, so $x$ is in the domain).
• Graph: Starts at the origin $(0, 0)$ and curves upwards and to the right. It is the upper half of the parabola $y^2 = x$.
• Properties: The function $f: [0, \infty) \to [0, \infty)$ is bijective.

(Imagine a graph of $y=\sqrt{x}$).

Familiarity with these standard functions, their properties, and their graphical shapes is crucial for success in mathematics involving real functions.

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