Writing Inequality Notation for Domain and Range

Introduction

Key Concepts

1. Understanding Domain and Range

In mathematics, the domain of a function refers to all possible input values (typically represented by 'x') that the function can accept without resulting in undefined or non-real outputs. Conversely, the range refers to all possible output values (typically represented by 'y') that a function can produce based on its domain.

For example, consider the function $f(x) = \sqrt{x}$. Here, the domain consists of all non-negative real numbers ($x \geq 0$) because taking the square root of a negative number is not defined in the set of real numbers. The range, in this case, is also all non-negative real numbers ($y \geq 0$) since the square root of any non-negative number is non-negative.

2. Importance of Inequality Notation

Inequality notation provides a concise and precise way to describe the domain and range of functions. It allows for clear communication of the boundaries within which a function operates, which is essential for solving equations, graphing functions, and understanding their behavior.

Using inequality notation helps in various applications, such as determining limits, analyzing function behavior, and solving real-world problems where constraints on inputs and outputs exist.

3. Expressing Domain Using Inequality

To express the domain of a function using inequality notation, identify all permissible values of 'x' that make the function defined. This involves considering factors such as:

• Denominators: Ensure that denominators are not zero, as division by zero is undefined.
• Radicals: Ensure that expressions under even roots (like square roots) are non-negative.
• Logarithms: Ensure that arguments of logarithmic functions are positive.

Example: Determine the domain of the function $g(x) = \frac{1}{x-3}$.

Solution: The denominator $x-3$ cannot be zero.

Therefore, the domain is all real numbers except $x = 3$, which can be expressed in inequality notation as:

4. Expressing Range Using Inequality

To express the range of a function using inequality notation, determine all possible 'y' values that correspond to the domain’s 'x' values. This typically involves solving the function for 'x' in terms of 'y' and identifying the possible 'y' values.

Example: Determine the range of the function $h(x) = 2x + 5$.

Solution: Since this is a linear function with a slope of 2, as 'x' takes any real value, 'y' will also take any real value. Thus, the range is:

Expressed in inequality notation:

5. Techniques for Writing Inequality Notation

Several techniques aid in determining appropriate inequality notation for domain and range:

• Set Builder Notation: Describes a set by stating the properties that its members must satisfy, e.g., $\{ x \ | \ x > 0 \}$.
• Interval Notation: Represents intervals on the real number line, e.g., $(0, \infty)$ for all positive real numbers.
• Inequality Symbols: Directly express conditions using <, ≤, >, and ≥.

For the purpose of this article, we'll focus on using inequality symbols to maintain clarity and compliance with the given instructions.

6. Common Scenarios and Examples

6.1. Polynomial Functions

Polynomial functions, such as $f(x) = x^2 - 4x + 3$, are defined for all real numbers. Thus, their domain and range can be determined by analyzing their graphs.

The domain is:

To find the range, consider the vertex of the parabola. Completing the square:

The vertex is at $(2, -1)$. Since the parabola opens upwards, the range is:

6.2. Rational Functions

Consider $g(x) = \frac{1}{x+2}$. The function is undefined when $x + 2 = 0$, i.e., $x = -2$. Thus, the domain is:

As 'x' approaches $-2$ from both sides, 'y' approaches infinity and negative infinity respectively, covering all real numbers except $y = 0$. Therefore, the range is:

6.3. Radical Functions

Take the function $h(x) = \sqrt{5 - x}$. The expression under the square root must be non-negative:

Thus, the domain is:

Since square roots yield non-negative results, the range is:

7. Graphical Interpretation

Graphing functions provides a visual representation of the domain and range. The domain is represented by the projection of the graph along the x-axis, while the range is the projection along the y-axis.

For instance, the graph of $f(x) = x^2$ is a parabola opening upwards with its vertex at the origin $(0,0)$. The domain is all real numbers:

And the range is all non-negative real numbers:

8. Special Functions and Their Domains and Ranges

Certain functions have specific domain and range characteristics:

• Exponential Functions: For $f(x) = a^x$ where $a > 0$, the domain is $-\infty < x < \infty$ and the range is $y > 0$.
• Logarithmic Functions: For $f(x) = \log_a(x)$ where $a > 0$, the domain is $x > 0$ and the range is $-\infty < y < \infty$.
• Trigonometric Functions: Functions like $\sin(x)$ and $\cos(x)$ have domains of all real numbers and ranges between -1 and 1.

9. Solving Inequalities to Find Domain and Range

Solving inequalities involves isolating the variable and determining the intervals that satisfy the inequality. This process is essential in defining the domain and range:

• For Domain: Identify restrictions on 'x' and solve the resulting inequalities.
• For Range: Manipulate the function to solve for 'y' in terms of 'x' and determine the inequalities that 'y' must satisfy.

Example: Find the domain and range of $f(x) = \frac{2}{\sqrt{4 - x}}$.

Solution:

Finding the Domain:

• The denominator $\sqrt{4 - x}$ must not be zero and the expression under the square root must be non-negative.

Thus, the domain is:

Finding the Range:

• Let $y = \frac{2}{\sqrt{4 - x}}$
• Since $\sqrt{4 - x} > 0$, $y$ is positive.
• As $x$ approaches $4$ from below, $\sqrt{4 - x}$ approaches $0$, making $y$ approach infinity.
• As $x$ approaches $-\infty$, $\sqrt{4 - x}$ becomes large, making $y$ approach $0$.

Therefore, the range is:

10. Common Mistakes to Avoid

When writing inequality notation for domain and range, students often make the following errors:

• Ignoring Restrictions: Failing to consider values that make the function undefined, such as division by zero or negative arguments under even roots.
• Incorrect Inequality Symbols: Using wrong inequality symbols, leading to incorrect domain or range.
• Overlooking Asymptotes: Not accounting for vertical or horizontal asymptotes in rational functions.
• Assuming All Functions Are Defined Everywhere: Assuming polynomial functions are defined for all real numbers without verifying.

Awareness of these common pitfalls ensures accurate determination of domain and range using inequality notation.

11. Practice Problems

To reinforce the concepts, here are some practice problems:

1. Find the domain and range of $f(x) = \frac{5}{x+1}$.
2. Determine the domain and range of $g(x) = \sqrt{9 - x^2}$.
3. Express the domain and range of $h(x) = \log(x - 2)$ using inequality notation.
4. Find the domain and range of $k(x) = \frac{x}{\sqrt{x - 3}}$.

Solutions:

1. Function: $f(x) = \frac{5}{x+1}$
   Domain: $x + 1 \neq 0$ $\Rightarrow$ $x \neq -1$
   

   Expressed as:

   $$ x < -1 \quad \text{or} \quad x > -1 $$ Range: All real numbers except $y = 0$.
   

   Expressed as:

   $$ y \neq 0 $$
2. Function: $g(x) = \sqrt{9 - x^2}$
   Domain: $9 - x^2 \geq 0$ $\Rightarrow$ $x^2 \leq 9$ $\Rightarrow$ $-3 \leq x \leq 3$
   

   Expressed as:

   $$ -3 \leq x \leq 3 $$ Range: Since square roots yield non-negative values and the maximum value of $9 - x^2$ is $9$, the range is:
   $$ 0 \leq y \leq 3 $$
3. Function: $h(x) = \log(x - 2)$
   Domain: $x - 2 > 0$ $\Rightarrow$ $x > 2$
   

   Expressed as:

   $$ x > 2 $$ Range: All real numbers.
   

   Expressed as:

   $$ -\infty < y < \infty $$
4. Function: $k(x) = \frac{x}{\sqrt{x - 3}}$
   Domain: $x - 3 > 0$ $\Rightarrow$ $x > 3$
   

   Expressed as:

   $$ x > 3 $$ Range: Since $x > 3$ and $\sqrt{x - 3} > 0$, $y$ can take any positive real value.
   

   Expressed as:

   $$ y > 0 $$

Comparison Table

Summary and Key Takeaways