https://file.aiquickdraw.com/mathbot/file/17258565930241lzfjx05.jpg

Based on this graph evaluate: g(f(-1))=___

From the image, the problem asks to evaluate \( g(f(-1)) \) based on the graph provided.

To solve this:
1. We first need to find \( f(-1) \) from the graph of \( f(x) \).
2. Then, we'll use this value to find \( g(f(-1)) \) by inputting the result into the graph of \( g(x) \).

Let's break it down:

1. **Evaluate \( f(-1) \)**: On the graph, trace the point where \( x = -1 \) on the function \( f(x) \) (blue line). Based on the graph, the \( y \)-value when \( x = -1 \) gives \( f(-1) \).

2. **Evaluate \( g(f(-1)) \)**: Once we have \( f(-1) \), we substitute it into the function \( g(x) \) (red line). Find the point on the \( g(x) \) graph where the \( x \)-value equals \( f(-1) \), and read the corresponding \( y \)-value to get \( g(f(-1)) \).

Let me check the graph and extract the values for you.From the graph:

1. **Finding \( f(-1) \)**: 
   When \( x = -1 \) on the function \( f(x) \) (blue line), the corresponding \( y \)-value appears to be \( 1 \). Therefore, \( f(-1) = 1 \).

2. **Finding \( g(f(-1)) \)**: 
   Now that we know \( f(-1) = 1 \), we use this value in the function \( g(x) \) (red line). Looking at the graph of \( g(x) \), when \( x = 1 \), the corresponding \( y \)-value appears to be \( -2 \). Therefore, \( g(1) = -2 \).

Thus, \( g(f(-1)) = -2 \).

Would you like more details or have any further questions? Here are some related questions to expand your understanding:

1. What is the domain and range of the functions \( f(x) \) and \( g(x) \) from the graph?
2. How do you evaluate a composition of functions in general?
3. What is the significance of the order in function composition?
4. Can \( g(f(x)) \) always be computed for any value of \( x \)? Why or why not?
5. What are common mistakes when interpreting composed function graphs?

**Tip:** When dealing with composition of functions, always evaluate the inner function first before proceeding to the outer function.

Learn how to evaluate the composition of functions g(f(x)) using graph interpretation. This example shows how to find g(f(-1)) by first locating f(-1) on the function graph, and then using that result to find g(f(-1)) on another graph. Suitable for students in Grades 9-12.

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