What Is the Domain and What Is the Range of a Function?

Discover the secrets of functions! What Is The Domain And What Is The Range? Unlock the full potential of mathematical functions with what.edu.vn’s comprehensive guide. Understand inputs, outputs, and relationships between them. Explore real-world applications and examples to master these essential concepts. Learn about function analysis, input values, output values, and mathematical relationships.

1. Understanding the Basics: What is the Domain and What is the Range?

The domain of a function is the complete set of possible values that you can put into a function. It’s the set of all independent variable values for which the function is defined. The range of a function is the complete set of all possible resulting values of the dependent variable (usually denoted as y, or f(x)), after we have substituted the domain. In simpler terms, the domain is all the possible x-values that will output a valid y-value, and the range is all the possible y-values you can get after plugging in all the x-values.

1.1. Formal Definitions of Domain and Range

Domain: Given a function f(x), the domain is the set of all x values for which f(x) is a real number. Mathematically, it can be expressed as:

Domain = {x ∈ ℝ | f(x) is real}
Range: The range is the set of all values that f(x) takes when x varies across its domain. Mathematically:

Range = {y ∈ ℝ | y = f(x) for some x in the domain}

1.2. Why Are Domain and Range Important?

Understanding the domain and range is crucial for several reasons:

Function Definition: They help to accurately define a function, ensuring it produces valid outputs for given inputs.
Graphing: They dictate the extent of a function’s graph, showing where the function exists on the coordinate plane.
Real-World Applications: In practical scenarios, they provide context. For example, if f(x) represents profit, the domain might represent the number of items sold, and the range represents the possible profit values.
Mathematical Operations: They are essential in calculus and analysis, influencing limits, continuity, and differentiability.

1.3. Domain and Range in the Context of Functions

The domain and range are fundamental aspects of functions. They help define the behavior and limitations of a function, which is why they are essential in various fields such as physics, engineering, economics, and computer science. Recognizing these parameters allows for accurate modeling and analysis in these domains.

2. Determining the Domain: How to Find Valid Input Values

Finding the domain of a function involves identifying all possible input values (x-values) that will produce a real number as an output. This often involves considering restrictions that might make the function undefined.

2.1. Common Restrictions on the Domain

Division by Zero: The denominator of a fraction cannot be zero. Therefore, any x-value that makes the denominator zero must be excluded from the domain.
Square Roots of Negative Numbers: In the realm of real numbers, you cannot take the square root (or any even root) of a negative number. Thus, the expression inside the square root must be greater than or equal to zero.
Logarithms of Non-Positive Numbers: Logarithms are undefined for zero and negative numbers. The argument of a logarithm must be strictly greater than zero.
Other Functions with Restrictions: Inverse trigonometric functions (like arcsin and arccos) have restricted domains because of their definitions.

2.2. Strategies for Finding the Domain

Identify Potential Restrictions: Look for any of the restrictions listed above (division by zero, even roots of negative numbers, logarithms of non-positive numbers, etc.).
Set Up Inequalities: Form inequalities based on these restrictions. For example, if you have a square root, set the expression inside the square root to be greater than or equal to zero.
Solve the Inequalities: Solve the inequalities to find the x-values that satisfy the conditions.
Write the Domain: Express the domain in interval notation, set notation, or graphically.

2.3. Examples of Finding the Domain

Example 1: Rational Function

Consider the function ( f(x) = frac{1}{x – 3} ).
The denominator cannot be zero, so ( x – 3 neq 0 ), which means ( x neq 3 ). Therefore, the domain is all real numbers except 3. In interval notation:
Domain: ( (-infty, 3) cup (3, infty) )

Example 2: Square Root Function

Consider the function ( g(x) = sqrt{2x + 4} ).
The expression inside the square root must be non-negative, so ( 2x + 4 geq 0 ). Solving for x:
[
2x geq -4
x geq -2
]
Therefore, the domain is all real numbers greater than or equal to -2. In interval notation:
Domain: ( [-2, infty) )

Example 3: Logarithmic Function

Consider the function ( h(x) = ln(x + 5) ).
The argument of the logarithm must be greater than zero, so ( x + 5 > 0 ). Solving for x:
[
x > -5
]
Therefore, the domain is all real numbers greater than -5. In interval notation:
Domain: ( (-5, infty) )

2.4. Advanced Techniques for Complex Functions

For more complex functions involving combinations of different types (e.g., rational functions with square roots), apply the restrictions sequentially and find the intersection of the resulting intervals. For instance, if you have ( k(x) = frac{sqrt{x + 2}}{x – 1} ), you need ( x + 2 geq 0 ) and ( x – 1 neq 0 ), leading to ( x geq -2 ) and ( x neq 1 ). Therefore, the domain is ( [-2, 1) cup (1, infty) ).

3. Determining the Range: How to Find All Possible Output Values

Finding the range of a function involves determining all possible output values (y-values or f(x) values) that the function can produce when x varies across its domain. This can be more challenging than finding the domain, as it often requires understanding the function’s behavior.

3.1. Methods for Finding the Range

Graphical Analysis:
- Plot the Function: Graph the function to visually inspect the range.
- Identify Minimum and Maximum: Determine the lowest and highest points on the graph to find the minimum and maximum y-values.
- Note Asymptotes: Observe any horizontal asymptotes, as these indicate bounds on the range.
Algebraic Analysis:
- Solve for x: If possible, solve the function ( y = f(x) ) for x in terms of y, i.e., find ( x = f^{-1}(y) ).
- Find the Domain of the Inverse: The domain of ( f^{-1}(y) ) is the range of ( f(x) ).
Calculus Techniques:
- Find Critical Points: Use derivatives to find critical points (where ( f'(x) = 0 ) or is undefined).
- Evaluate at Critical Points and Endpoints: Evaluate the function at these points to find local maxima and minima, which help determine the range.
Understanding Function Behavior:
- Consider End Behavior: Examine what happens to ( f(x) ) as ( x ) approaches positive and negative infinity.
- Analyze Transformations: Understand how transformations (shifts, stretches, reflections) affect the range.

3.2. Examples of Finding the Range

Example 1: Linear Function

Consider the function ( f(x) = 2x + 3 ).
Since there are no restrictions on x, the domain is all real numbers. As x varies across all real numbers, ( 2x + 3 ) also varies across all real numbers. Therefore, the range is all real numbers.
Range: ( (-infty, infty) )

Example 2: Quadratic Function

Consider the function ( g(x) = x^2 – 4x + 5 ).
To find the range, complete the square:
[
g(x) = (x – 2)^2 + 1
]
Since ( (x – 2)^2 ) is always non-negative, the minimum value of ( g(x) ) is 1 (when ( x = 2 )). There is no upper bound, so the range is all real numbers greater than or equal to 1.
Range: ( [1, infty) )

Example 3: Rational Function with Horizontal Asymptote

Consider the function ( h(x) = frac{1}{x} ).
The domain is all real numbers except 0. As ( x ) approaches infinity, ( h(x) ) approaches 0. However, ( h(x) ) never actually equals 0. The range is all real numbers except 0.
Range: ( (-infty, 0) cup (0, infty) )

Example 4: Square Root Function

Consider the function ( k(x) = sqrt{x} ).
The domain is ( [0, infty) ). As ( x ) varies from 0 to infinity, ( sqrt{x} ) also varies from 0 to infinity. Therefore, the range is all non-negative real numbers.
Range: ( [0, infty) )

3.3. Advanced Techniques and Considerations

Piecewise Functions: For piecewise functions, find the range of each piece separately and then combine them.
Trigonometric Functions: Understand the ranges of trigonometric functions. For example, the range of ( sin(x) ) and ( cos(x) ) is ( [-1, 1] ).
Composition of Functions: When dealing with composite functions, find the range of the inner function first, then use that as the domain for the outer function.

4. Domain and Range of Common Functions

Understanding the domain and range of common functions can serve as a foundation for analyzing more complex functions. Below is a table summarizing the domain and range of several basic functions:

Function	Equation	Domain	Range
Linear Function	( f(x) = mx + b )	( (-infty, infty) )	( (-infty, infty) )
Quadratic Function	( f(x) = ax^2 + bx + c )	( (-infty, infty) )	( [k, infty) ) or ( (-infty, k] ), where ( k ) is the vertex’s y-coordinate
Polynomial Function	( f(x) = a_nx^n + … + a_0 )	( (-infty, infty) )	Depends on degree and coefficients; odd degree: ( (-infty, infty) ), even degree: bounded on one side
Rational Function	( f(x) = frac{P(x)}{Q(x)} )	All real numbers except where ( Q(x) = 0 )	Depends on the specific function, often involves asymptotes
Square Root Function	( f(x) = sqrt{x} )	( [0, infty) )	( [0, infty) )
Exponential Function	( f(x) = a^x )	( (-infty, infty) )	( (0, infty) )
Logarithmic Function	( f(x) = log_a(x) )	( (0, infty) )	( (-infty, infty) )
Sine Function	( f(x) = sin(x) )	( (-infty, infty) )	( [-1, 1] )
Cosine Function	( f(x) = cos(x) )	( (-infty, infty) )	( [-1, 1] )
Tangent Function	( f(x) = tan(x) )	All real numbers except ( x = frac{(2n+1)pi}{2} )	( (-infty, infty) )

4.1. Linear Functions

Linear functions are of the form ( f(x) = mx + b ), where ( m ) and ( b ) are constants.

Domain: Since you can plug in any real number for ( x ), the domain is all real numbers, ( (-infty, infty) ).
Range: Unless ( m = 0 ) (in which case it’s a horizontal line), the range is also all real numbers, ( (-infty, infty) ). If ( m = 0 ), the range is just ( {b} ).

4.2. Quadratic Functions

Quadratic functions are of the form ( f(x) = ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are constants.

Domain: You can plug in any real number for ( x ), so the domain is all real numbers, ( (-infty, infty) ).
Range: The range depends on the vertex of the parabola and whether the parabola opens upwards (if ( a > 0 )) or downwards (if ( a < 0 )). The vertex’s y-coordinate gives the minimum or maximum value of the function. If ( a > 0 ), the range is ( [k, infty) ), where ( k ) is the y-coordinate of the vertex. If ( a < 0 ), the range is ( (-infty, k] ).

4.3. Polynomial Functions

Polynomial functions are sums of terms, each of which is a constant times a power of ( x ).

Domain: For any polynomial, the domain is all real numbers, ( (-infty, infty) ).
Range: The range depends on the degree of the polynomial. For odd-degree polynomials, the range is all real numbers, ( (-infty, infty) ). For even-degree polynomials, the range is bounded on one side (either ( [k, infty) ) or ( (-infty, k] ) for some ( k )).

4.4. Rational Functions

Rational functions are of the form ( f(x) = frac{P(x)}{Q(x)} ), where ( P(x) ) and ( Q(x) ) are polynomials.

Domain: The domain is all real numbers except where the denominator ( Q(x) = 0 ).
Range: The range can be more complicated. It often involves considering horizontal asymptotes and the function’s behavior near vertical asymptotes.

4.5. Square Root Functions

Square root functions are of the form ( f(x) = sqrt{x} ).

Domain: Since you can only take the square root of non-negative numbers, the domain is ( [0, infty) ).
Range: The range is also ( [0, infty) ), as the square root of a non-negative number is always non-negative.

4.6. Exponential Functions

Exponential functions are of the form ( f(x) = a^x ), where ( a ) is a constant.

Domain: The domain is all real numbers, ( (-infty, infty) ).
Range: The range is ( (0, infty) ), as ( a^x ) is always positive.

4.7. Logarithmic Functions

Logarithmic functions are of the form ( f(x) = log_a(x) ).

Domain: Since you can only take the logarithm of positive numbers, the domain is ( (0, infty) ).
Range: The range is all real numbers, ( (-infty, infty) ).

4.8. Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent.

Sine and Cosine Functions: For ( f(x) = sin(x) ) and ( f(x) = cos(x) ):
- Domain: All real numbers, ( (-infty, infty) ).
- Range: ( [-1, 1] ).
Tangent Function: For ( f(x) = tan(x) ):
- Domain: All real numbers except ( x = frac{(2n+1)pi}{2} ) where ( n ) is an integer (because cosine is zero at these points, and tangent is sine divided by cosine).
- Range: All real numbers, ( (-infty, infty) ).

5. Domain and Range in Real-World Applications

Domain and range aren’t just abstract mathematical concepts; they’re useful in understanding real-world phenomena. Let’s examine some applications:

5.1. Physics: Projectile Motion

In physics, projectile motion describes the path of an object thrown into the air. The height ( h(t) ) of the object at time ( t ) can be modeled by a quadratic function:
[ h(t) = -gt^2 + v_0t + h_0 ]
where ( g ) is the acceleration due to gravity, ( v_0 ) is the initial vertical velocity, and ( h_0 ) is the initial height.

Domain: The domain for this function is typically ( t geq 0 ) because time cannot be negative. Depending on the context, the domain might also be bounded by the time when the object hits the ground (i.e., ( h(t) = 0 )).
Range: The range represents the height of the projectile. The maximum height can be found by determining the vertex of the quadratic function. The range would then be from the initial height up to the maximum height.

5.2. Economics: Cost Functions

In economics, a cost function ( C(x) ) describes the total cost of producing ( x ) units of a product.

Domain: The domain is the number of units produced, which must be non-negative integers. So, the domain is typically ( x geq 0 ), where ( x ) is an integer.
Range: The range is the total cost, which depends on the production level. If there are fixed costs, the range will start from that fixed cost and increase as production increases.

5.3. Biology: Population Growth

Exponential growth models are used to describe population growth. The population ( P(t) ) at time ( t ) can be modeled by:
[ P(t) = P_0e^{kt} ]
where ( P_0 ) is the initial population, ( e ) is the base of the natural logarithm, and ( k ) is the growth rate.

Domain: The domain is time ( t ), which is non-negative, so ( t geq 0 ).
Range: The range is the population size, which starts from ( P_0 ) and increases exponentially. So, the range is ( [P_0, infty) ).

5.4. Computer Science: Algorithm Analysis

In computer science, the time complexity of an algorithm describes how the runtime of the algorithm grows as the input size increases. This is often expressed using Big O notation.

Domain: The domain is the input size ( n ), which is a positive integer, so ( n geq 1 ).
Range: The range is the runtime of the algorithm, which depends on the algorithm’s complexity. For example, if an algorithm has a time complexity of ( O(n^2) ), the range would be the set of possible runtime values as ( n ) varies.

5.5. Everyday Life: Temperature Conversion

Consider converting temperature from Celsius to Fahrenheit using the formula:
[ F = frac{9}{5}C + 32 ]

Domain: The domain could be any possible Celsius temperature. However, in a practical context, it might be limited to a reasonable range of temperatures for a particular environment.
Range: The range is the corresponding Fahrenheit temperatures. If the Celsius temperature is limited, the Fahrenheit temperature will also be limited.

5.6. Practical Implications

In each of these examples, understanding the domain and range helps in interpreting the results and ensuring they make sense within the context of the problem. It provides bounds on possible values and helps avoid nonsensical results (e.g., negative time or infinite costs).

6. How to Graph Functions and Identify Domain and Range Visually

Graphing functions is an invaluable tool for understanding their behavior, including identifying the domain and range visually. A graph provides a clear picture of the function’s inputs and outputs, making it easier to determine these essential characteristics.

6.1. Basic Graphing Techniques

Create a Table of Values: Choose a set of x-values, calculate the corresponding y-values, and plot the points on a coordinate plane.
Plot Key Points: Identify and plot important points such as intercepts (where the graph crosses the x-axis and y-axis), vertices (for quadratic functions), and any points of discontinuity.
Connect the Points: Draw a smooth curve or line through the plotted points, keeping in mind the function’s general shape and any asymptotes.
Use Graphing Tools: Utilize graphing calculators or online tools like Desmos or GeoGebra for accurate and dynamic visualizations.

6.2. Identifying the Domain from a Graph

The domain of a function can be determined by observing the x-values for which the function is defined on the graph.

Look for Boundaries: Check for any vertical lines (vertical asymptotes) or closed/open circles that indicate where the function is undefined.
Project onto the x-axis: Imagine projecting the entire graph onto the x-axis. The portion of the x-axis that is covered represents the domain of the function.
Interval Notation: Express the domain in interval notation, considering any breaks or endpoints.

6.3. Identifying the Range from a Graph

The range of a function can be determined by observing the y-values that the function attains on the graph.

Look for Boundaries: Check for any horizontal lines (horizontal asymptotes) or highest and lowest points on the graph.
Project onto the y-axis: Imagine projecting the entire graph onto the y-axis. The portion of the y-axis that is covered represents the range of the function.
Interval Notation: Express the range in interval notation, considering any endpoints or gaps.

6.4. Examples of Graphing and Identifying Domain and Range

Example 1: Linear Function ( f(x) = 2x + 1 )

Graph: A straight line with a slope of 2 and a y-intercept of 1.
Domain: The graph extends infinitely in both x-directions. Therefore, the domain is ( (-infty, infty) ).
Range: The graph extends infinitely in both y-directions. Therefore, the range is ( (-infty, infty) ).

Example 2: Quadratic Function ( g(x) = x^2 – 4 )

Graph: A parabola opening upwards with its vertex at (0, -4).
Domain: The graph extends infinitely in both x-directions. Therefore, the domain is ( (-infty, infty) ).
Range: The lowest point on the graph is -4, and it extends upwards infinitely. Therefore, the range is ( [-4, infty) ).

Example 3: Rational Function ( h(x) = frac{1}{x – 2} )

Graph: A hyperbola with a vertical asymptote at ( x = 2 ) and a horizontal asymptote at ( y = 0 ).
Domain: The graph is defined for all x-values except ( x = 2 ). Therefore, the domain is ( (-infty, 2) cup (2, infty) ).
Range: The graph takes on all y-values except ( y = 0 ). Therefore, the range is ( (-infty, 0) cup (0, infty) ).

Example 4: Square Root Function ( k(x) = sqrt{x + 3} )

Graph: A curve that starts at ( x = -3 ) and increases to the right.
Domain: The graph is defined for ( x geq -3 ). Therefore, the domain is ( [-3, infty) ).
Range: The graph starts at ( y = 0 ) and increases upwards infinitely. Therefore, the range is ( [0, infty) ).

6.5. Using Technology

Graphing calculators and online tools like Desmos and GeoGebra can be particularly helpful for visualizing more complex functions and accurately determining their domain and range. These tools allow you to zoom in and out, trace the graph, and identify key points and asymptotes more easily.

6.6. Importance of Visual Analysis

Visual analysis through graphing provides a comprehensive understanding of a function’s behavior, making it easier to determine its domain and range. This method is especially useful for functions with complex equations or those that are difficult to analyze algebraically.

7. Domain and Range of Trigonometric Functions

Trigonometric functions are fundamental in mathematics and have specific domains and ranges that are essential to understand for various applications.

7.1. Sine Function (( f(x) = sin(x) ))

Domain: The sine function is defined for all real numbers. You can input any angle (in radians or degrees) into the sine function, so the domain is:
[ (-infty, infty) ]
Range: The sine function oscillates between -1 and 1. The output values are always within this interval:
[ [-1, 1] ]
This means that for any ( x ), ( -1 leq sin(x) leq 1 ).

7.2. Cosine Function (( f(x) = cos(x) ))

Domain: Like the sine function, the cosine function is defined for all real numbers:
[ (-infty, infty) ]
Range: The cosine function also oscillates between -1 and 1:
[ [-1, 1] ]
For any ( x ), ( -1 leq cos(x) leq 1 ).

7.3. Tangent Function (( f(x) = tan(x) ))

Domain: The tangent function is defined as ( tan(x) = frac{sin(x)}{cos(x)} ). It is undefined where ( cos(x) = 0 ). This occurs at ( x = frac{(2n + 1)pi}{2} ), where ( n ) is an integer. Therefore, the domain is all real numbers except these points:
[ x neq frac{(2n + 1)pi}{2}, quad n in mathbb{Z} ]
In interval notation, the domain is a union of intervals:
[ bigcup_{n = -infty}^{infty} left( frac{(2n – 1)pi}{2}, frac{(2n + 1)pi}{2} right) ]
Range: The tangent function can take any real value. As ( x ) approaches ( frac{(2n + 1)pi}{2} ), ( tan(x) ) approaches ( infty ) or ( -infty ). Therefore, the range is:
[ (-infty, infty) ]

7.4. Cosecant Function (( f(x) = csc(x) ))

Domain: The cosecant function is defined as ( csc(x) = frac{1}{sin(x)} ). It is undefined where ( sin(x) = 0 ). This occurs at ( x = npi ), where ( n ) is an integer. The domain is all real numbers except these points:
[ x neq npi, quad n in mathbb{Z} ]
In interval notation:
[ bigcup_{n = -infty}^{infty} (npi, (n + 1)pi) ]
Range: The cosecant function is always greater than or equal to 1 or less than or equal to -1. It never takes values between -1 and 1:
[ (-infty, -1] cup [1, infty) ]

7.5. Secant Function (( f(x) = sec(x) ))

Domain: The secant function is defined as ( sec(x) = frac{1}{cos(x)} ). It is undefined where ( cos(x) = 0 ), which occurs at ( x = frac{(2n + 1)pi}{2} ), where ( n ) is an integer. The domain is:
[ x neq frac{(2n + 1)pi}{2}, quad n in mathbb{Z} ]
In interval notation:
[ bigcup_{n = -infty}^{infty} left( frac{(2n – 1)pi}{2}, frac{(2n + 1)pi}{2} right) ]
Range: Like the cosecant function, the secant function is always greater than or equal to 1 or less than or equal to -1:
[ (-infty, -1] cup [1, infty) ]

7.6. Cotangent Function (( f(x) = cot(x) ))

Domain: The cotangent function is defined as ( cot(x) = frac{cos(x)}{sin(x)} ). It is undefined where ( sin(x) = 0 ), which occurs at ( x = npi ), where ( n ) is an integer. The domain is:
[ x neq npi, quad n in mathbb{Z} ]
In interval notation:
[ bigcup_{n = -infty}^{infty} (npi, (n + 1)pi) ]
Range: The cotangent function can take any real value:
[ (-infty, infty) ]

7.7. Summary Table

Function	Domain	Range
( sin(x) )	( (-infty, infty) )	( [-1, 1] )
( cos(x) )	( (-infty, infty) )	( [-1, 1] )
( tan(x) )	( x neq frac{(2n + 1)pi}{2}, quad n in mathbb{Z} )	( (-infty, infty) )
( csc(x) )	( x neq npi, quad n in mathbb{Z} )	( (-infty, -1] cup [1, infty) )
( sec(x) )	( x neq frac{(2n + 1)pi}{2}, quad n in mathbb{Z} )	( (-infty, -1] cup [1, infty) )
( cot(x) )	( x neq npi, quad n in mathbb{Z} )	( (-infty, infty) )

7.8. Understanding Trigonometric Functions

Understanding the domain and range of trigonometric functions is critical for solving equations, graphing, and applying these functions in various fields like physics, engineering, and signal processing.

8. Transforming Functions and Their Impact on Domain and Range

Transforming functions can significantly alter their graphs, affecting both the domain and range. Understanding how different transformations impact these properties is essential for analyzing and manipulating functions effectively.

8.1. Types of Transformations

Vertical Shifts:
- Transformation: ( f(x) rightarrow f(x) + c )
- Effect: Shifts the graph upward by ( c ) units if ( c > 0 ) and downward by ( |c| ) units if ( c < 0 ).
- Impact on Domain: No change.
- Impact on Range: The range shifts by ( c ) units.
Horizontal Shifts:
- Transformation: ( f(x) rightarrow f(x – c) )
- Effect: Shifts the graph to the right by ( c ) units if ( c > 0 ) and to the left by ( |c| ) units if ( c < 0 ).
- Impact on Domain: The domain shifts by ( c ) units.
- Impact on Range: No change.
Vertical Stretches and Compressions:
- Transformation: ( f(x) rightarrow c cdot f(x) )
- Effect: Stretches the graph vertically by a factor of ( c ) if ( |c| > 1 ) and compresses it if ( 0 < |c| < 1 ). If ( c < 0 ), it also reflects the graph across the x-axis.
- Impact on Domain: No change.
- Impact on Range: The range is multiplied by ( c ).
Horizontal Stretches and Compressions:
- Transformation: ( f(x) rightarrow f(cx) )
- Effect: Compresses the graph horizontally by a factor of ( |c| ) if ( |c| > 1 ) and stretches it if ( 0 < |c| < 1 ). If ( c < 0 ), it also reflects the graph across the y-axis.
- Impact on Domain: The domain is divided by ( c ).
- Impact on Range: No change.
Reflections:
- Across the x-axis: ( f(x) rightarrow -f(x) )
  - Impact on Domain: No change.
  - Impact on Range: The range is multiplied by -1.
- Across the y-axis: ( f(x) rightarrow f(-x) )
  - Impact on Domain: The domain is multiplied by -1.
  - Impact on Range: No change.

8.2. Examples of Transformations and Their Effects

Example 1: Vertical Shift

Let ( f(x) = x^2 ) with a domain and range of ( (-infty, infty) ) and ( [0, infty) ) respectively.
Consider ( g(x) = x^2 + 3 ). This shifts the graph of ( f(x) ) upward by 3 units.

Domain of ( g(x) ): ( (-infty, infty) ) (no change)
Range of ( g(x) ): ( [3, infty) ) (shifted upward by 3 units)

Example 2: Horizontal Shift

Let ( f(