Find the domain and the range of the following real functions: (a) 𝑓(𝑥) = (𝑥2 − 4)/(x+2)

(f) 𝑓(𝑥)=|𝑥|−𝑥 /(2𝑥)

(g)𝑓(𝑥)= 𝑥−3/( 2𝑥 + 1)

(h) 𝑓(𝑥) = 1 /(1−𝑥2)

(i) 𝑓(𝑥) = √𝑥2 − 4

(j) 𝑓(𝑥)= 𝑥+2/( |𝑥 + 2|)

Let's analyze each function step by step to find the **domain** (the set of all possible input values) and the **range** (the set of all possible output values).

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### (a) \( f(x) = \frac{x^2 - 4}{x + 2} \)

#### Domain:
- The denominator \( x + 2 \) must not be zero, so \( x \neq -2 \).
- Thus, the domain is:
\[
\boxed{(-\infty, -2) \cup (-2, \infty)}
\]

#### Simplification:
- The numerator \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \), so the function simplifies:
  \[
  f(x) = \frac{(x - 2)(x + 2)}{x + 2}
  \]
  For \( x \neq -2 \), this simplifies to:
  \[
  f(x) = x - 2
  \]
  
#### Range:
- Since \( f(x) = x - 2 \) for all \( x \neq -2 \), the range is all real numbers:
\[
\boxed{(-\infty, \infty)}
\]

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### (f) \( f(x) = \frac{|x| - x}{2x} \)

#### Domain:
- The denominator \( 2x \) must not be zero, so \( x \neq 0 \).
- Thus, the domain is:
\[
\boxed{(-\infty, 0) \cup (0, \infty)}
\]

#### Simplification:
- For \( x > 0 \), \( |x| = x \), so \( f(x) = \frac{x - x}{2x} = 0 \).
- For \( x < 0 \), \( |x| = -x \), so \( f(x) = \frac{-x - x}{2x} = \frac{-2x}{2x} = -1 \).

#### Range:
- For \( x > 0 \), \( f(x) = 0 \).
- For \( x < 0 \), \( f(x) = -1 \).
- Therefore, the range is:
\[
\boxed{\{-1, 0\}}
\]

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### (g) \( f(x) = \frac{x - 3}{2x + 1} \)

#### Domain:
- The denominator \( 2x + 1 \neq 0 \), so \( x \neq -\frac{1}{2} \).
- Thus, the domain is:
\[
\boxed{(-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)}
\]

#### Range:
- \( f(x

Find the domain and the range of the following real functions: (a) f(x) = (x^2 − 4)/(x + 2), (f) f(x) = |x| − x / (2x), (g) f(x) = (x − 3)/(2x + 1), (h) f(x) = 1/(1 − x^2), (i) f(x) = √(x^2 − 4), (j) f(x) = (x + 2)/|x + 2|

Explore step-by-step solutions to find the domain and range of functions such as f(x) = (x^2 − 4)/(x + 2) and others involving absolute values and rational expressions. Learn how to handle square roots, denominators, and more. Suitable for Grades 10-12.

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