Refer To Functions S And T. Find The Indicated Function And Write The Domain In Interval Notation. Write

Refer To Functions S And T. Find The Indicated Function And Write The Domain In Interval Notation. Write

Understanding how to analyze functions, determine their expressions, and identify their domains is fundamental in mathematics, especially in calculus and algebra. When dealing with composite functions or functions derived from others, it’s crucial to carefully interpret the given information, find the explicit form of the function, and then determine its domain. This article provides a comprehensive guide to working with functions referred to as S and T, including how to find their explicit forms based on given conditions and how to express their domains using interval notation.

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Understanding Functions and Their Notation

Before diving into specific functions, it’s essential to grasp some basic concepts:

What is a Function?

A function is a relation between a set of inputs (domain) and a set of outputs (range) where each input corresponds to exactly one output. Functions are often denoted by letters such as f, g, S, T, etc.

Function Notation

Standard notation is f(x), where f is the function and x is the input. When functions are named S and T, they might be described explicitly or implicitly through operations, compositions, or given formulas.

Domain and Range

• Domain: The set of all possible input values for the function.
• Range: The set of all possible output values.

Expressing the domain in interval notation makes it easier to communicate the extent of the function’s inputs visually and mathematically.

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Common Types of Functions and Their Domains

When analyzing functions like S and T, you may encounter various types, such as:

• Polynomial functions


• Rational functions


• Square root functions


• Composite functions

Each type has specific considerations for its domain:

Polynomial Functions

• Typically defined for all real numbers.
• Domain: \((-\infty, \infty)\)

Rational Functions

• Defined for all real numbers except where the denominator is zero.
• To find the domain, set the denominator ≠ 0.

Square Root and Radical Functions

• Defined where the expression inside the radical is ≥ 0.
• To find the domain, solve the inequality inside the radical.

Composite Functions

• Formed by combining functions, such as \( (f \circ g)(x) = f(g(x)) \).
• Domain depends on the domain of g and the range of g fitting into the domain of f.

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Step-by-Step Approach to Find Functions S and T and Their Domains

Let’s consider a typical problem where functions S and T are defined via certain relations or compositions, and the goal is:

1. To find the explicit form of the indicated function.


2. To determine the domain of that function in interval notation.

Step 1: Understand the Given Information

• Identify how functions S and T are defined.


• Note any given formulas, compositions, or relations.


• Recognize the type of the functions involved.

Step 2: Derive the Explicit Form

• Use algebraic operations, substitutions, or inverse functions if necessary.


• Simplify expressions to obtain a clear formula for the function.

Step 3: Determine the Domain

• Analyze the explicit formula.


• Identify any restrictions (denominators, radicals, logs).


• Solve inequalities or equations to find the set of permissible input values.


• Express this set in interval notation.

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Example of Finding a Function and Its Domain

Suppose you are given:

• \( S(x) = \sqrt{x + 3} \)


• \( T(x) = \frac{1}{x - 2} \)

And you are asked to find:

• The composite function \( R(x) = S(T(x)) \)


• The domain of \( R(x) \) in interval notation.

Solution:

Step 1: Find \( R(x) \)

• \( R(x) = S(T(x)) = \sqrt{T(x) + 3} \)

Step 2: Substitute \( T(x) \)

• \( R(x) = \sqrt{\frac{1}{x - 2} + 3} \)

Step 3: Simplify the expression inside the radical

• \( R(x) = \sqrt{\frac{1 + 3(x - 2)}{x - 2}} \)


• \( R(x) = \sqrt{\frac{1 + 3x - 6}{x - 2}} = \sqrt{\frac{3x - 5}{x - 2}} \)

Step 4: Find the domain

• The expression inside the radical must be ≥ 0:

\[
\frac{3x - 5}{x - 2} \geq 0
\]

• Also, the denominator cannot be zero: \( x \neq 2 \).

Step 5: Solve the inequality

• Critical points are where numerator or denominator is zero:


• Numerator zero: \( 3x - 5 = 0 \Rightarrow x = \frac{5}{3} \)


• Denominator zero: \( x = 2 \)


• Sign analysis:

| Interval | Test Point | Sign of numerator \(3x - 5\) | Sign of denominator \(x - 2\) | Fraction \(\frac{3x - 5}{x - 2}\) | Result (≥ 0) |
|------------|--------------|------------------------------|------------------------------|----------------------------|--------------|
| \(-\infty, \frac{5}{3}\) | 0 | \( -5 \) (negative) | \( -2 \) (negative) | positive | Yes |
| \(\frac{5}{3}, 2\) | 1.5 | \( 4.5 - 5 = -0.5 \) (negative) | \( -0.5 \) (negative) | positive | Yes |
| \(2, \infty\) | 3 | \( 9 - 5 = 4 \) (positive) | \( 1 \) (positive) | positive | Yes |

• But note that at \( x = 2 \), the function is undefined due to division by zero.


• At \( x = \frac{5}{3} \), the radical equals zero, which is acceptable.

Final domain:

• From \( -\infty \) up to but not including 2, since the denominator zero point is excluded.


• At \( x = \frac{5}{3} \), the expression inside the radical is zero, so it's included.


• From just greater than \( \frac{5}{3} \) up to 2, the expression stays non-negative.


• After 2, the expression is again positive, but since 2 is excluded, the domain is:

\[
(-\infty, 2) \setminus \{2\} \quad \text{and} \quad \left[\frac{5}{3}, 2\right)
\]

Expressed in interval notation:

\[
(-\infty, 2) \quad \text{excluding } x=2, \quad \text{but including } \frac{5}{3}
\]

This simplifies to:

\[
(-\infty, 2) \quad \text{with } x \neq 2
\]

Because at \( x = \frac{5}{3} \), the radical is zero, so included.

Therefore, the domain:

\[
(-\infty, 2) \setminus \{2\}
\]

or in interval notation:

\[
(-\infty, 2)
\]

with the understanding that \( x \neq 2 \).

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General Tips for Finding Functions and Domains

• Always check for restrictions: Denominator zero, radical expression negative, or logarithm of non-positive number.
• Solve inequalities carefully: Use test points and sign charts.
• Express the domain accurately: Use interval notation, including parentheses for excluding points and brackets for inclusive points.
• Verify the function's definition: After deriving the explicit form, test points in the domain to ensure correctness.

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Summary

In solving problems involving functions S and T, the core steps involve:

1. Interpreting the given information to understand how the functions are defined.


2. Deriving the explicit formula for the indicated function, often involving substitution, algebraic manipulation, or inverse functions.


3. Determining the domain by analyzing restrictions such as division by zero, radicals, or logs, and solving inequalities.


4. Expressing the domain in interval notation, carefully considering the points of restriction or inclusion.

Mastering these steps enhances your ability to analyze complex functions, understand their behavior, and communicate their properties effectively.

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Conclusion

Understanding how to find functions like S and T and their domains is essential for advanced mathematics and real-world applications. Whether dealing with simple algebraic functions, radicals, or compositions, a systematic approach ensures accuracy and clarity. Always pay attention to restrictions imposed by the function's form, and use interval notation to succinctly express the domain. With practice, analyzing functions becomes more intuitive, enabling you to solve more complex problems confidently.

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