Question

if (f(x)=f(g(x))), where (f(-1)=2), (f(-1)=3), (f(4)=3), (g(4)= - 1), and (g(4)=4), find (f(4)). (f(4)=)

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: Substitute $x = 4$

When $x = 4$, we have $F^{\prime}(4)=f^{\prime}(g(4))\cdot g^{\prime}(4)$.

Step3: Use given values

We know that $g(4)=-1$ and $g^{\prime}(4)=4$. So $F^{\prime}(4)=f^{\prime}(-1)\cdot4$.

Step4: Substitute $f^{\prime}(-1)$ value

Since $f^{\prime}(-1)=3$, then $F^{\prime}(4)=3\times4$.

Answer:

$12$