11. -/1 points details my notes if f(x)=f(g(x)), where f(0)=4, f(0)=7, f(-1)=3, g(-1)=0, and g(-1)=3, find…

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $F(x)=f(g(x))$, then $F^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$.

Step2: Evaluate at $x = - 1$

Substitute $x=-1$ into the chain - rule formula. We get $F^{\prime}(-1)=f^{\prime}(g(-1))\cdot g^{\prime}(-1)$.

Step3: Use given values

We know that $g(-1) = 0$ and $g^{\prime}(-1)=3$, and $f^{\prime}(0)=7$. Since $g(-1) = 0$, then $f^{\prime}(g(-1))=f^{\prime}(0)$. So $F^{\prime}(-1)=f^{\prime}(0)\cdot g^{\prime}(-1)$.

Step4: Calculate the result

Substitute $f^{\prime}(0) = 7$ and $g^{\prime}(-1)=3$ into the equation. We have $F^{\prime}(-1)=7\times3$. $F^{\prime}(-1)=21$

Answer:

$21$