Chapter 1, Problem 88RE

a.

To determine

Expression of $fog$

Answer to Problem 88RE

  $fog=x+3$

Explanation of Solution

Given:

Functions are given as;

  $f\left(x\right)=x^2-4,g\left(x\right)=\sqrt[3]{x+7}$

  $\begin{array}{l}fog=f\left(g\left(x\right)\right)\\ =f\left(\sqrt[3]{x+7}\right)\\ ={\left(\sqrt[3]{x+7}\right)}^3-4\\ ={\left(x+7\right)}^{3\times \frac{1}{3}}-4\\ =x+3\end{array}$

b.

To determine

Expression of $gof$ and domain of both composite function.

Answer to Problem 88RE

  $gof=\sqrt[3]{x+7}$ and domain of both the composite function will be $\left(-\infty ,\infty \right)$

Explanation of Solution

Given:

Functions are given as;

  $f\left(x\right)=x^2-4,g\left(x\right)=\sqrt[3]{x+7}$

  $\begin{array}{l}gof=g\left(f\left(x\right)\right)\\ =g\left(x^3-4\right)\\ =\sqrt[3]{x^3-4+7}\\ =\sqrt[x]{x^3+7}\end{array}$

Domain of any function is defined as a set of real number for which function give real value.

Now, for $fog=x+3$

Domain of this function is all real number

Therefore domain will be $\left(-\infty ,\infty \right)$

Now, for $gof=\sqrt[3]{x+7}$

As it is known that cube root is defined for both positive and negative value therefore domain of this function will be also all real number.

Therefore domain will be $\left(-\infty ,\infty \right)$