Domain and Range of Rational and Radical Functions

Understanding the Domain

The domain of a function is the set of all possible input values (usually \(x\)) for which the function is defined. This involves:

1. Avoiding division by zero in rational functions.
2. Ensuring the radicand (expression under a square root or even root) is non-negative in radical functions.
3. Combining the above when both situations are present.

Understanding the Range

The range of a function is the set of all possible output values. Determining the range often involves:

1. Analyzing the behavior of the function at extremes (\(x \to \infty\) or \(x \to -\infty\)).
2. Checking specific restrictions imposed by the function.

Worked Examples

1. Rational Functions with Linear Numerator and Denominator

Consider the rational function: \[ f(x) = \frac{2x+3}{x-5}. \]

Domain:

• The denominator \(x - 5\) cannot be zero.
• Solve \(x - 5 = 0\): \(x = 5\).
• Domain: \(x \in \mathbb{R}, x \neq 5\), or \((-\infty, 5) \cup (5, \infty)\).

Range:

• Write \(f(x)\) in terms of \(y\): \(y = \frac{2x+3}{x-5}\).
• Rearrange to express \(x\) in terms of \(y\): \[ y(x-5) = 2x+3 \implies yx - 5y = 2x + 3 \implies x(y-2) = 5y + 3. \]
• Solve for \(x\): \[ x = \frac{5y + 3}{y-2}. \]
• For the range, the denominator \(y - 2 \neq 0\), so \(y \neq 2\).
• Range: \(y \in \mathbb{R}, y \neq 2\), or \((-\infty, 2) \cup (2, \infty)\).

2. Rational Function with Quadratic Denominator

Consider: \[ f(x) = \frac{x+2}{x^2-4}. \]

Domain:

• Factor the denominator: \(x^2 - 4 = (x-2)(x+2)\).
• Denominator cannot be zero: \(x - 2 = 0 \; \text{or} \; x + 2 = 0 \implies x = 2, -2\).
• Domain: \(x \in \mathbb{R}, x \neq -2, 2\), or \((-\infty, -2) \cup (-2, 2) \cup (2, \infty)\).

Range:

• Rearrange \(y = \frac{x+2}{x^2-4}\) to express \(x\) in terms of \(y\): \[ y(x^2-4) = x+2 \implies yx^2 - x - 4y - 2 = 0. \]
• This is a quadratic equation in \(x\): \[ yx^2 - x - 4y - 2 = 0. \]
• Discriminant: \[ \Delta = (-1)^2 - 4(y)(-4y-2) = 1 + 16y^2 + 8y. \]
• \(\Delta > 0\) for all \(y\), so \(f(x)\) has no range restrictions.
• Range: \(\mathbb{R}\).

3. Rational Function with Complex Terms

Example: \[ f(x) = 8 + \frac{x+3}{2x-7}. \]

Domain:

• Denominator \(2x-7 = 0 \implies x = \frac{7}{2}\).
• Domain: \(x \in \mathbb{R}, x \neq \frac{7}{2}\).

Range:

• Express \(y = 8 + \frac{x+3}{2x-7}\).
• Rearrange: \[ y - 8 = \frac{x+3}{2x-7}. \]
• Simplify: \[ x(2y - 16 - 1) = 7(y - 8) + 3. \]
• Since \(y \neq 8\), the range is \(\mathbb{R} \setminus \{8\}\).

4. Rational Function with a Quadratic and Radical Combination

Consider: \[ f(x) = \frac{x}{\sqrt{x^2+7}}. \]

Domain:

• The denominator \(\sqrt{x^2+7}\) is always defined as the radicand \(x^2+7 \geq 0\) (true for all \(x\)).
• Domain: \(x \in \mathbb{R}\).

Range:

• Analyze behavior as \(x \to \infty\) or \(x \to -\infty\): \[ f(x) \to \pm 1. \]
• Check \(f(x) = 0\): possible.
• Range: \((-1, 1)\).

5. Rational Function with Radical Denominator

Consider: \[ f(x) = \frac{1}{\sqrt{x+4}}. \]

Domain:

• The denominator is defined when \(x+4 > 0\), so \(x > -4\).
• Domain: \((-4, \infty)\).

Range:

• As \(x \to -4^+\), \(f(x) \to \infty\).
• As \(x \to \infty\), \(f(x) \to 0^+\).
• Range: \((0, \infty)\).

6. Rational Function with Radical Numerator and Quadratic Denominator

Consider: \[ f(x) = \frac{\sqrt{x+2}}{x^2-1}. \]

Domain:

• The numerator \(\sqrt{x+2}\) requires \(x+2 \geq 0\), so \(x \geq -2\).
• The denominator \(x^2-1=0\) when \(x=\pm 1\).
• Combine restrictions: \(x \geq -2\) and \(x \neq \pm 1\).
• Domain: \([ -2, -1) \cup (-1, 1) \cup (1, \infty)\).

Range:

• Analyze behavior as \(x \to \infty\): \(f(x) \to 0^+\).
• Near vertical asymptotes (\(x = \pm 1\)), \(f(x) \to \pm \infty\).
• As \(x \to -2^+\), \(f(x) \to 0^+\).
• Range: \((0, \infty)\).

7. Rational Function with Denominator Always Positive

Consider: \[ f(x) = \frac{x+1}{x^2+1}. \]

Domain:

• The denominator \(x^2+1 > 0\) for all \(x \in \mathbb{R}\).
• Domain: \(x \in \mathbb{R}\).

Range:

• Analyze as \(x \to \pm \infty\): \(f(x) \to 0^+\).
• Critical points occur where \(f'(x)=0\).
• Minimum and maximum values occur, but \(f(x)\) never crosses \(0\) or becomes unbounded.
• Range: \((-\infty, \infty)\).

8. Rational Function with Radical Numerator and Positive Quadratic Denominator

Consider: \[ f(x) = \frac{\sqrt{x+1}}{x^2+5}. \]

Domain:

• The numerator \(\sqrt{x+1}\) requires \(x+1 \geq 0\), so \(x \geq -1\).
• The denominator \(x^2+5 > 0\) for all \(x \in \mathbb{R}\).
• Domain: \([ -1, \infty)\).

Range:

• As \(x \to -1^+\), \(f(x) \to 0^+\).
• As \(x \to \infty\), \(f(x) \to 0^+\).
• Maximum values occur at critical points.
• Range: \((0, M]\), where \(M\) is the maximum value of \(f(x)\).

The maximum value of \[ f(x) = \frac{\sqrt{x+1}}{x^2 + 5} \] occurs at approximately \[ x = 0.786, \] and the maximum value is: \[ M \approx 0.238. \] Thus, the range of the function is: \[ (0, 0.238]. \]

Practice Problems

1. Find the domain and range of: \[ f(x) = \frac{\sqrt{x-3}}{x^2+4}. \]
2. Analyze the domain and range of: \[ f(x) = \frac{x}{x^2+1}. \]
3. For the function \[ f(x) = \frac{\sqrt{x+5}}{x^2-9}, \] determine the domain and range.
4. Find the domain and range of: \[ f(x) = \frac{x-2}{x^2+1}. \]
5. Determine the domain and range of: \[ f(x) = \frac{\sqrt{x}}{x^2+1}. \]
6. Find the domain and range of: \[ f(x) = \frac{x+1}{x-3}. \]
7. Analyze the domain and range of: \[ f(x) = \frac{2}{x^2+1}. \]
8. For the function \[ f(x) = \frac{x-2}{\sqrt{x+6}}, \] determine the domain and range.
9. Analyze the domain and range of: \[ f(x) = \frac{5x+3}{x^2-9}. \]
10. Find the domain and range of: \[ f(x) = \frac{1}{\sqrt{x^2+4}}. \]

Answers to Practice Problems

1. Domain: \(x \geq 3\); Range: \((0, \infty)\).
2. Domain: \(\mathbb{R}\); Range: \((-\infty, \infty)\).
3. Domain: \(x \geq -5, x \neq \pm 3\); Range: \((0, \infty)\).
4. Domain: \(\mathbb{R}\); Range: \((-\infty, \infty)\).
5. Domain: \(x \geq 0\); Range: \((0, \infty)\).
6. Domain: \(x \neq 3\); Range: \(y \neq 1\).
7. Domain: \(\mathbb{R}\); Range: \((0, 2]\).
8. Domain: \(x \geq -6\); Range: \((-\infty, 0) \cup (2, \infty)\).
9. Domain: \(x \neq \pm 3\); Range: All real numbers except horizontal asymptotes.
10. Domain: \(\mathbb{R}\); Range: \((0, 0.5]\).