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 $x$:

  \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!