Use the figures above, which show the functions f(x) and g(x), to find the following values.
1. f(g(1))=
2. g(f(−1))=
3. g(g(−2))=
To find the values of the given expressions, let's go step by step:
1. To find f(g(1)), we need to first find g(1) and then substitute that value into f(x). Looking at the graph of g(x), we can see that g(1) corresponds to the value of the y-coordinate when x = 1. By tracing a vertical line from x = 1 to the graph of g(x), we can see that g(1) = 3.
Next, we substitute g(1) into f(x). By tracing a horizontal line from y = 3 to the graph of f(x), we can determine the corresponding x-value. It appears to be between x = 0 and x = 1, closer to x = 0. Let's estimate it as x ≈ 0.2.
Therefore, f(g(1)) ≈ f(3) ≈ 0.2.
2. To find g(f(-1)), we start by finding f(-1) and then substitute that value into g(x). By tracing a vertical line from x = -1 to the graph of f(x), we see that f(-1) = -2.
Next, we substitute f(-1) into g(x). By tracing a horizontal line from y = -2 to the graph of g(x), we can find the corresponding x-value. It appears to be between x = -3 and x = -2, closer to x = -2. Let's estimate it as x ≈ -2.7.
Therefore, g(f(-1)) ≈ g(-2) ≈ -2.7.
3. To find g(g(-2)), we need to evaluate g(-2) first and then substitute that value back into g(x). By tracing a vertical line from x = -2 to the graph of g(x), we find that g(-2) = -3.
Next, we substitute g(-2) into g(x). By tracing a horizontal line from y = -3 to the graph of g(x), we can find the corresponding x-value. It appears to be approximately x = 0.
Therefore, g(g(-2)) ≈ g(-3) ≈ 0.
Please note that the values provided are approximations based on visual estimation from the graph. For more precise calculations, we would need the actual equations or more accurate data.
To find the values for the given expressions, we need to substitute the appropriate values into the functions and then evaluate them step by step. Let's start with the first expression:
1. f(g(1)):
To find g(1), we look for the value of g(x) when x = 1. From the figure, g(1) = -2.
Next, we substitute this result into f(x) to find f(g(1)). So, we need to find f(-2). From the figure, f(-2) = 4.
Therefore, f(g(1)) = f(-2) = 4.
2. g(f(-1)):
To find f(-1), we need to find the value of f(x) when x = -1. From the figure, f(-1) = 2.
Now, we substitute this value into g(x) to find g(f(-1)). So, we need to find g(2). From the figure, g(2) = 1.
Therefore, g(f(-1)) = g(2) = 1.
3. g(g(-2)):
To find g(-2), we need to find the value of g(x) when x = -2. From the figure, g(-2) = 3.
Now, we substitute this value into g(x) again to find g(g(-2)). So, we need to find g(3). From the figure, g(3) = 0.
Therefore, g(g(-2)) = g(3) = 0.
To summarize:
1. f(g(1)) = 4
2. g(f(-1)) = 1
3. g(g(-2)) = 0
1. f(g(1))=2
2. g(f(-1))=-2
3. g(g(-2))=-3