Questions: Use f(x)=3x-2 and g(x)=5-x^2 to evaluate the expression. (a) f(f(2)) (b) g(g(3))

Transcript text: Use $f(x)=3 x-2$ and $g(x)=5-x^{2}$ to evaluate the expression. (a) $f(f(2))$ (b) $\quad g(g(3))$

Solution

To solve the given problems, we need to evaluate the functions $ f(x) $ and $ g(x) $ using the provided expressions.

(a) For $ f(f(2)) $, first calculate $ f(2) $ using the function $ f(x) = 3x - 2 $. Then, use the result to find $ f $ of that result.

(b) For $ g(g(3)) $, first calculate $ g(3) $ using the function $ g(x) = 5 - x^2 $. Then, use the result to find $ g $ of that result.

Step 1: Evaluate $ f(2) $

To find $ f(f(2)) $, we first need to evaluate $ f(2) $ using the function $ f(x) = 3x - 2 $.

\[ f(2) = 3(2) - 2 = 6 - 2 = 4 \]

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

Now, use the result from Step 1 to find $ f(f(2)) $.

\[ f(f(2)) = f(4) = 3(4) - 2 = 12 - 2 = 10 \]

Step 3: Evaluate $ g(3) $

To find $ g(g(3)) $, we first need to evaluate $ g(3) $ using the function $ g(x) = 5 - x^2 $.

\[ g(3) = 5 - 3^2 = 5 - 9 = -4 \]

Step 4: Evaluate $ g(g(3)) $

Now, use the result from Step 3 to find $ g(g(3)) $.

\[ g(g(3)) = g(-4) = 5 - (-4)^2 = 5 - 16 = -11 \]

$\boxed{f(f(2)) = 10}$

$\boxed{g(g(3)) = -11}$

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