The task is to find the maximal domain and the range of the following functions, expressed as compositions of simpler functions:

(a) $f(x) = \frac{x^2}{x^2 + 1}$

Domain:

This function is a rational function. Since the denominator $x^2 + 1$ is always positive for all real $x$, there are no restrictions on $x$. Therefore, the domain is: $D(f) = \mathbb{R}$ (all real numbers).

Range:

Let's analyze the behavior of $f(x) = \frac{x^2}{x^2 + 1}$:

• As $x \to \infty$, $f(x) \to 1$.
• As $x \to 0$, $f(0) = 0$.
• $f(x)$ is symmetric about the y-axis (even function), and its value increases as $|x|$ increases from 0 to infinity.

Since $0 \leq f(x) < 1$ for all $x \in \mathbb{R}$, the range is: $R(f) = [0, 1)$

(b) $f(x) = \sqrt{x(4 - x)}$

Domain:

To ensure the function is real, the expression under the square root must be non-negative: $x(4 - x) \geq 0$ This is a quadratic inequality. The roots of the equation $x(4 - x) = 0$ are $x = 0$ and $x = 4$. The quadratic opens downwards, meaning $x(4 - x) \geq 0$ for $0 \leq x \leq 4$. Therefore, the domain is: $D(f) = [0, 4]$

Range:

The expression inside the square root is a parabola with its maximum at $x = 2$, where: $f(2) = \sqrt{2(4 - 2)} = \sqrt{4} = 2$

• At $x = 0$, $f(0) = \sqrt{0} = 0$.
• At $x = 4$, $f(4) = \sqrt{0} = 0$.

Thus, the range of the function is: $R(f) = [0, 2]$

Would you like further details on these solutions, or have any specific questions?

Related Questions:

1. How would the domain and range change if the function in part (a) were modified to $\frac{x^3}{x^2 + 1}$?
2. Can you graph the function $f(x) = \frac{x^2}{x^2 + 1}$ and describe its critical points?
3. What is the effect of adding a constant inside the square root in part (b), like $\sqrt{x(4 - x) + 1}$?
4. How can you find the inverse function for $f(x) = \frac{x^2}{x^2 + 1}$, if it exists?
5. If you shift the quadratic inside the square root to $\sqrt{x(5 - x)}$, how does that affect the domain and range?

Tip:

Always check for restrictions in the domain by analyzing the denominator (in rational functions) or expressions under square roots and logarithms.