Questions: Evaluate f(g(1)) and g(f(2)), if possible. If not type NA. f(x) = x / (2 - x), g(x) = 4 - x^2 f(g(1)) = g(f(2)) =

Transcript text: Evaluate $f(g(1))$ and $g(f(2))$, if possible. If not type NA. \[ f(x)=\frac{x}{2-x}, g(x)=4-x^{2} \] $f(g(1))=$ $\square$ $g(f(2))=$ $\square$

Solution

Solution Steps

To solve the problem of evaluating $ f(g(1)) $ and $ g(f(2)) $, we need to follow these steps:

Evaluate $ g(1) $: Substitute $ x = 1 $ into the function $ g(x) = 4 - x^2 $ to find $ g(1) $.
Evaluate $ f(g(1)) $: Use the result from step 1 and substitute it into the function $ f(x) = \frac{x}{2-x} $.
Evaluate $ f(2) $: Substitute $ x = 2 $ into the function $ f(x) = \frac{x}{2-x} $ to find $ f(2) $.
Evaluate $ g(f(2)) $: Use the result from step 3 and substitute it into the function $ g(x) = 4 - x^2 $.

Step 1: Evaluate $ g(1) $

First, we need to evaluate $ g(1) $ using the function $ g(x) = 4 - x^2 $.

\[ g(1) = 4 - (1)^2 = 4 - 1 = 3 \]

Step 2: Evaluate $ f(g(1)) $

Now that we have $ g(1) = 3 $, we can evaluate $ f(g(1)) = f(3) $ using the function $ f(x) = \frac{x}{2-x} $.

\[ f(3) = \frac{3}{2-3} = \frac{3}{-1} = -3 \]

Step 3: Evaluate $ f(2) $

Next, we need to evaluate $ f(2) $ using the function $ f(x) = \frac{x}{2-x} $.

\[ f(2) = \frac{2}{2-2} = \frac{2}{0} \]

Since division by zero is undefined, $ f(2) $ is not defined.

Step 4: Evaluate $ g(f(2)) $

Since $ f(2) $ is not defined, $ g(f(2)) $ is also not defined. Therefore, we write NA for $ g(f(2)) $.