let (f(x)=x^{3}) and (g(x)=\frac{x}{x - 1}). if (h) is the function defined by (h(x)=f(g(x))), which of the…

let (f(x)=x^{3}) and (g(x)=\frac{x}{x - 1}). if (h) is the function defined by (h(x)=f(g(x))), which of the following gives a correct expression for (h(x))? a. (3(g(x))^{2}(\frac{x}{x - 1})) b. (3(g(x))^{2}=3(\frac{x}{x - 1})^{2}) c. (3(g(x))^{2}g(x)=3(\frac{x}{x - 1})^{2}(\frac{x}{x - 1})) d. ((g(x))^{3}=(\frac{x}{x - 1})^{3})

let (f(x)=x^{3}) and (g(x)=\frac{x}{x - 1}). if (h) is the function defined by (h(x)=f(g(x))), which of the following gives a correct expression for (h(x))? a. (3(g(x))^{2}(\frac{x}{x - 1})) b. (3(g(x))^{2}=3(\frac{x}{x - 1})^{2}) c. (3(g(x))^{2}g(x)=3(\frac{x}{x - 1})^{2}(\frac{x}{x - 1})) d. ((g(x))^{3}=(\frac{x}{x - 1})^{3})

Answer

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $h(x)=f(g(x))$, then $h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. Given $f(x)=x^{3}$, then $f^{\prime}(x) = 3x^{2}$. Given $g(x)=\frac{x}{x - 1}$, using the quotient - rule $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime}v - uv^{\prime}}{v^{2}}$ where $u = x$, $u^{\prime}=1$, $v=x - 1$, $v^{\prime}=1$, we have $g^{\prime}(x)=\frac{1\cdot(x - 1)-x\cdot1}{(x - 1)^{2}}=\frac{x - 1 - x}{(x - 1)^{2}}=-\frac{1}{(x - 1)^{2}}$.

Step2: Apply chain - rule

$h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)$. Substitute $g(x)$ into $f^{\prime}(x)$: $f^{\prime}(g(x)) = 3(g(x))^{2}=3\left(\frac{x}{x - 1}\right)^{2}$. Then $h^{\prime}(x)=3\left(\frac{x}{x - 1}\right)^{2}\cdot\left(-\frac{1}{(x - 1)^{2}}\right)$.

Answer:

B. $3(g(x))^{2}=3\left(\frac{x}{x - 1}\right)^{2}$