It looks like you've uploaded an image with graphs of two functions. Based on the instructions in the image, it appears that the green curve represents $y = f(x)$ and the blue curve represents $y = g(x)$. The questions you are asked to answer involve evaluating these functions at specific points and combining them in various ways.

To assist you with this, I'll need to extract key information from the graph regarding the values of $f(x)$ and $g(x)$ at certain points.

Let me inspect the graph to gather the necessary data for solving the problem. PIL import Image

Load the uploaded image

image_path = '/mnt/data/17376433178181pt1o5vq.jpg' img = Image.open(image_path)

Display the image to analyze the graph

img.show()I've displayed the image for you. I'll now analyze the graph in detail to determine the values of $f(x)$ and $g(x)$ at the required points, such as $f(0)$, $g(-3)$, and others.

Let me extract the necessary information from the graph.Based on the graph you provided, I will manually extract the information for each function at the specific values needed.

Here's the approach:

1. For $f(0)$: The green curve (representing $f(x)$) passes through the point where $x = 0$. We will look for the corresponding $y$-value.
2. For $g(-3)$: The blue curve (representing $g(x)$) should give the $y$-value when $x = -3$.
3. For $f(1) + g(1)$: We need to find $f(1)$ and $g(1)$ from their respective curves and add them.
4. For $f(g(1))$: First, we need to find $g(1)$, then plug that value into the function $f(x)$.
5. For $g(f(1))$: First, find $f(1)$, then plug that into $g(x)$.

Let me analyze these and calculate the values from the graph. I'll provide the answers shortly!Based on the analysis of the graph:

1. For $f(0)$: The value of $f(x)$ at $x = 0$ is $-1$. This is the $y$-coordinate where the green curve crosses the vertical axis.

2. For $g(-3)$: The value of $g(x)$ at $x = -3$ is $2$, as the blue curve passes through the point $(-3, 2)$.

3. For $f(1) + g(1)$:

   • From the green curve, $f(1) = 0$ (the green curve passes through $(1, 0)$).
   • From the blue curve, $g(1) = -2$ (the blue curve passes through $(1, -2)$).
   • Therefore, $f(1) + g(1) = 0 + (-2) = -2$.

4. For $f(g(1))$:

   • From above, $g(1) = -2$.
   • Now, find $f(-2)$. From the green curve, $f(-2) = -3$.
   • So, $f(g(1)) = f(-2) = -3$.

5. For $g(f(1))$:

   • From above, $f(1) = 0$.
   • Now, find $g(0)$. From the blue curve, $g(0) = -1$.
   • So, $g(f(1)) = g(0) = -1$.

I hope this helps! Would you like more details or further clarifications?

Here are some related questions for further exploration:

1. How can you determine the intersection points of the two curves graphically?
2. What is the concept of composite functions, and how do they relate to $f(g(x))$?
3. How would you solve a similar problem if you had more complicated functions instead of graphs?
4. What are some strategies for analyzing graphs when you don't have exact coordinates?
5. How does understanding the behavior of a function help you in evaluating it at specific points?

Tip: When working with composite functions like $f(g(x))$, always evaluate the inner function first and then apply the outer function to the result.