Questions: If (f(x)=-2 x-5) and (g(x)=x^2+1), find (f(g(x))).

Transcript text: If $f(x)=-2 x-5$ and $g(x)=x^{\wedge} 2+1$, find $f(g(x))$.

Solution

Step 1: Understand the Problem

We are given two functions, $ f(x) = -2x - 5 $ and $ g(x) = x^2 + 1 $. We need to find the composition of these functions, specifically $ f(g(x)) $.

Step 2: Substitute $ g(x) $ into $ f(x) $

To find $ f(g(x)) $, we substitute $ g(x) = x^2 + 1 $ into $ f(x) $. This means we replace every $ x $ in $ f(x) $ with $ g(x) $.

\[ f(g(x)) = f(x^2 + 1) = -2(x^2 + 1) - 5 \]

Step 3: Simplify the Expression

Now, we simplify the expression:

\[ f(g(x)) = -2(x^2 + 1) - 5 = -2x^2 - 2 - 5 \]

Combine the constant terms:

\[ f(g(x)) = -2x^2 - 7 \]

The expression for $ f(g(x)) $ is $-2x^2 - 7$. Therefore, the correct multiple-choice answer is:

\[ \boxed{-2x^2 - 7} \]

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