if (f(x)=f(g(x))), where (f(-1) = 4), (f(-1)=7), (f(2)=3), (g(2)= - 1), and (g(2)=2), find (f(2)). (f(2)=)…
if (f(x)=f(g(x))), where (f(-1) = 4), (f(-1)=7), (f(2)=3), (g(2)= - 1), and (g(2)=2), find (f(2)). (f(2)=) - 4 (×)
Answer
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 $x = 2$
Substitute $x = 2$ into the chain - rule formula. We get $F^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)$.
Step3: Use given values
We know that $g(2)=-1$ and $g^{\prime}(2)=2$, and $f^{\prime}(-1) = 7$. So $F^{\prime}(2)=f^{\prime}(g(2))\cdot g^{\prime}(2)=f^{\prime}(-1)\cdot2$.
Step4: Calculate the result
Since $f^{\prime}(-1)=7$, then $F^{\prime}(2)=7\times2 = 14$.
Answer:
$14$