Let's determine the domain and range of the given piecewise function:

1. Domain:

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

For the given piecewise function:

• $f(x) = x^2 - 5$ is defined for $x \leq 0$.
• $f(x) = 2$ is defined for $0 < x < 3$.
• $f(x) = x + 1$ is defined for $x \geq 3$.

The function covers all values of $x$ from negative infinity to positive infinity without any gaps.

Domain: $(-\infty, \infty)$ or all real numbers $\mathbb{R}$.

2. Range:

The range of a function is the set of all possible output values (y-values) that the function can take.

Let's analyze each part of the function to determine the range:

• For $x^2 - 5$ when $x \leq 0$:

  • The minimum value occurs at $x = 0$, giving $f(0) = -5$.
  • As $x$ decreases (becomes more negative), $x^2$ increases, so $f(x)$ becomes more positive.
  • Thus, the range for this part is $(-5, \infty)$.

• For $f(x) = 2$ when $0 < x < 3$:

  • The output is constantly $2$, so the range for this part is just $\{2\}$.

• For $f(x) = x + 1$ when $x \geq 3$:

  • The smallest value occurs at $x = 3$, giving $f(3) = 4$.
  • As $x$ increases, $f(x)$ also increases.
  • Thus, the range for this part is $[4, \infty)$.

Now, combining the ranges of each piece:

Range: The combined range is $(-5, 2] \cup [4, \infty)$.

This indicates the function takes values starting from just above $-5$ up to $2$, and then from $4$ onward to infinity.

Do you want to delve into more details or have any questions?

Here are 5 related questions:

1. How would the domain and range change if the interval for $f(x) = x^2 - 5$ were expanded beyond $x = 0$?
2. Can you identify the points on the graph where the function is discontinuous?
3. What would be the range if $f(x) = 2$ were extended to include $x = 3$?
4. How does changing the coefficient of $x$ in the linear part $f(x) = x + 1$ affect the range?
5. What is the effect on the range if we alter the quadratic function to $f(x) = -(x^2) + 5$ for $x \leq 0$?

Tip: To determine the range of a piecewise function, evaluate the output values (y-values) over each interval separately, and then combine the results.