Find the domain and range of the following:
(a). g(x) = x^2/x+x^2

Let's analyze the function \( g(x) = \frac{x^2}{x + x^2} \) to determine its domain and range.

### 1. **Domain**:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

The function \( g(x) = \frac{x^2}{x + x^2} \) is a rational function, which means it is defined as long as the denominator is not zero.

- The denominator of \( g(x) \) is \( x + x^2 \). We need to find where this expression is equal to zero because the function is undefined at those points:
  \[
  x + x^2 = 0
  \]
  Factor the expression:
  \[
  x(1 + x) = 0
  \]
  This gives us two solutions:
  \[
  x = 0 \quad \text{or} \quad 1 + x = 0 \quad \Rightarrow \quad x = -1
  \]
  
Thus, the function is undefined at \( x = 0 \) and \( x = -1 \). The domain of \( g(x) \) excludes these points.

So, the domain is:
\[
\text{Domain of } g(x): \, \mathbb{R} \setminus \{-1, 0\}
\]
This means \( g(x) \) is defined for all real numbers except \( x = -1 \) and \( x = 0 \).

### 2. **Range**:
The range of a function is the set of all possible output values (y-values).

To find the range, let's analyze the behavior of the function.

- First, simplify the function if possible:
  \[
  g(x) = \frac{x^2}{x + x^2} = \frac{x^2}{x(1 + x)} = \frac{x}{1 + x}, \quad x \neq 0, -1
  \]
  The simplified form is \( g(x) = \frac{x}{1 + x} \).

- Next, find the possible values of \( y = g(x) \) by solving for \( x \) in terms of \( y \):
  \[
  y = \frac{x}{1 + x}
  \]
  Multiply both sides by \( 1 + x \):
  \[
  y(1 + x) = x
  \]
  Distribute \( y \):
  \[
  y + yx = x
  \]
  Rearrange to solve for \( x \):
  \[
  y + yx - x = 0
  \]
  Factor out \( x \):
  \[
  y + x(y - 1) = 0 \quad \Rightarrow \quad x(y - 1) = -y
  \quad \Rightarrow \quad x = \frac{-y}{y - 1}, \quad y \neq 1
  \]

  For \( y = 1 \), check the function directly:
  \[
  1 = \frac{x}{1 + x}
  \]
  \[
  1 + x = x
  \]
  This leads to a contradiction, so \( y = 1 \) is not in the range.

- \( g(x) \) can take all values except \( y = 1 \).

Thus, the range of \( g(x) \) is:
\[
\text{Range of } g(x): \, \mathbb{R} \setminus \{1\}
\]

### Summary:
- **Domain**: \( \mathbb{R} \setminus \{-1, 0\} \)
- **Range**: \( \mathbb{R} \setminus \{1\} \)

Would you like more details or have any questions?

Here are some related questions to consider:
1. How would the range change if the function were \( g(x) = \frac{x^2}{x + 2x^2} \)?
2. What happens to the domain if the function was \( g(x) = \frac{x^2 + 1}{x + x^2} \)?
3. What are the domain and range of the function \( g(x) = \frac{1}{x^2 + x} \)?
4. How does the function \( g(x) = \frac{x^2 + x}{x + 1} \) behave at \( x = -1 \)?
5. Can the range of a rational function be all real numbers? If so, under what conditions?

**Tip:** Simplifying the function first can often make finding the domain and range much easier!

Explore how to find the domain and range of the function g(x) = x^2/(x + x^2) with a detailed step-by-step solution. Understand the concept of rational functions and their domain exclusions. Suitable for students in Grades 11-12.

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