Let's work through this step by step based on the image provided.

(a) Finding the Domain:

The function is defined in three parts:

• $f(x) = \frac{2}{x^2}$ if $0 \leq x \leq 4$
• $f(x) = x + 12$ if $4 < x \leq 7$
• $f(x) = 19$ if $x > 7$

The domain consists of all $x$-values where the function is defined.

• For $f(x) = \frac{2}{x^2}$, it is defined for $0 \leq x \leq 4$.
• For $f(x) = x + 12$, it's defined for $4 < x \leq 7$.
• For $f(x) = 19$, it's defined for $x > 7$.

Thus, the domain is $[0, \infty)$.

(b) Finding Intercepts:

• Y-intercept: The y-intercept occurs where $x = 0$. Using the first part of the function: $f(0) = \frac{2}{0^2} = \infty$ This implies there is no y-intercept, because $f(0)$ is undefined.

• X-intercept: The x-intercept occurs where $f(x) = 0$.

  1. For $f(x) = \frac{2}{x^2}$, it can never equal zero.
  2. For $f(x) = x + 12$, set $x + 12 = 0$: $x = -12$ But this is not in the domain of the function.
  3. For $f(x) = 19$, it can never equal zero.

Thus, there is no x-intercept.

(c) Graphing the Function:

You need to select the correct graph based on the function. Here's how the function behaves in the three intervals:

1. For $0 \leq x \leq 4$, the function $f(x) = \frac{2}{x^2}$ decreases as $x$ increases.
2. For $4 < x \leq 7$, the function $f(x) = x + 12$ is a straight line with slope 1.
3. For $x > 7$, the function is constant, $f(x) = 19$.

Based on this behavior, the correct graph will reflect these characteristics.

(d) Finding the Range:

• For $0 \leq x \leq 4$, $f(x) = \frac{2}{x^2}$ decreases from $\frac{2}{4^2} = \frac{2}{16}$ to $\infty$.
• For $4 < x \leq 7$, $f(x) = x + 12$, so it increases from 16 to 19.
• For $x > 7$, $f(x) = 19$.

Thus, the range is $(0, 19]$.

Would you like any further details on any part of this?

Here are five related questions to explore:

1. What are the conditions for the existence of intercepts in different types of functions?
2. How do piecewise functions affect continuity at boundary points?
3. What are typical methods to find the range of piecewise functions?
4. How can graphing help in visualizing and confirming function properties?
5. What happens if the domain of a function includes a point where it's undefined?

Tip: For piecewise functions, always check the behavior at the boundaries, as they can sometimes indicate discontinuities or breaks.