Let's tackle each part of this problem step-by-step.

Part (a) - Graphing the Function and Its Inverse

To sketch a one-to-one function containing the points (-2, -2), (0, 4), and (1, 7):

1. Plot the points on a coordinate plane.
2. Draw a curve or line that passes through these points, ensuring the graph does not fold back on itself (to maintain the one-to-one property).
3. Once the function is plotted, reflect each point across the line $y = x$ to obtain the points for the inverse function. The points of the inverse will be (-2, -2), (4, 0), and (7, 1).
4. Draw the inverse by connecting these reflected points smoothly.

In comparing this with others' graphs, you might observe that the function and its inverse are symmetric with respect to the line $y = x$. This symmetry is a fundamental property of a function and its inverse.

Part (b) - Finding $f(g(x))$ and $g(f(x))$

Given: $f(x) = 3x + 4$ $g(x) = \frac{1}{3}(x - 4)$

To find $f(g(x))$:

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(\frac{1}{3}(x - 4)\right)$
2. Substitute into $f$: $= 3\left(\frac{1}{3}(x - 4)\right) + 4$
3. Simplify: $= (x - 4) + 4 = x$

To find $g(f(x))$:

1. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(3x + 4)$
2. Substitute into $g$: $= \frac{1}{3}((3x + 4) - 4)$
3. Simplify: $= \frac{1}{3}(3x) = x$

Thus, $f(g(x)) = x$ and $g(f(x)) = x$, which shows that $f$ and $g$ "undo" each other.

Part (c) - Are $f(x) = 3x + 4$ and $g(x) = \frac{1}{3}(x - 4)$ Inverses?

Yes, $f(x) = 3x + 4$ and $g(x) = \frac{1}{3}(x - 4)$ are inverses of each other. We demonstrated this in part (b) by showing that $f(g(x)) = x$ and $g(f(x)) = x$. This mutual cancellation is the defining property of inverse functions.

Part (d) - Graphing $f(x) = 3x + 4$, $g(x) = \frac{1}{3}(x - 4)$, and $y = x$

To graph these functions:

1. Plot $f(x) = 3x + 4$, a line with slope 3 and y-intercept 4.
2. Plot $g(x) = \frac{1}{3}(x - 4)$, a line with slope $\frac{1}{3}$ and y-intercept $-\frac{4}{3}$.
3. Plot $y = x$, a line that serves as the line of symmetry between $f$ and $g$.

The graphs of $f$ and $g$ will be reflections of each other across the line $y = x$, reinforcing that they are inverse functions. This visual symmetry is a characteristic of any function and its inverse.

Would you like to see a detailed graph or need further explanations?

Here are some questions to deepen understanding:

1. Why does a function need to be one-to-one to have an inverse function?
2. How can we confirm that a function is one-to-one?
3. What role does the line $y = x$ play in the concept of inverses?
4. How can you determine the inverse of a given function algebraically?
5. Can functions that are not one-to-one ever have inverses?

Tip: When verifying inverse functions, always check if $f(g(x)) = x$ and $g(f(x)) = x$; this confirms they "undo" each other.