The image provides three functions, and you are asked to determine their natural domain and range.

Let's analyze each function:

a. $f(x) = x^2 + 3$

• Domain: Since this is a polynomial function (a quadratic), its domain is all real numbers because any real number can be squared and added to 3.

  • Domain: $(-\infty, \infty)$

• Range: The minimum value of $x^2$ is 0, so the minimum value of $f(x) = x^2 + 3$ is 3. Since $x^2$ grows without bound as $|x|$ increases, the range is $[3, \infty)$.

  • Range: $[3, \infty)$

b. $f(x) = 1 + \sqrt{3 - x}$

• Domain: The square root function requires that the argument $3 - x \geq 0$, so: $3 - x \geq 0 \implies x \leq 3$

  • Domain: $(-\infty, 3]$

• Range: The smallest value of $\sqrt{3 - x}$ occurs when $x = 3$, giving $f(3) = 1 + \sqrt{0} = 1$. As $x$ approaches negative infinity, $\sqrt{3 - x}$ becomes arbitrarily large, so the range is $[1, \infty)$.

  • Range: $[1, \infty)$

c. $f(x) = \frac{x + 1}{(x - 3)(x + 1)}$

• Domain: The denominator must not be zero, so $(x - 3)(x + 1) \neq 0$. This occurs when $x \neq 3$ and $x \neq -1$.

  • Domain: $(-\infty, -1) \cup (-1, 3) \cup (3, \infty)$

• Range: To analyze the range, we notice that the function has vertical asymptotes at $x = -1$ and $x = 3$. There is no horizontal asymptote since the degree of the numerator equals the degree of the denominator, meaning the range is all real numbers except for specific values that are "skipped" by the asymptotes. A detailed exploration would confirm that the range is all real numbers.

  • Range: $(-\infty, \infty)$

Would you like further clarification or additional details?

Related Questions:

1. How do you determine the domain of a square root function?
2. What happens to the range when a function has a vertical asymptote?
3. Why do quadratic functions always have a domain of all real numbers?
4. Can you explain the difference between the domain and range in rational functions?
5. What are some common methods to find the range of more complicated functions?

Tip: When finding the range of a function, consider both the behavior as $x \to \infty$ and the specific values that might restrict the range (e.g., square roots, denominators).