Q.34 - Precalculus Enhanced with Graphing Utilities

Question

For the given functions f and g, find:  

$a) f\circ g b) g\circ f c) f \circ f d) g\circ g$

State the domain of each composite function. 

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{Part} \left(\mathrm{a}\right) f\circ g=4x^4+12x^2+10 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{b}\right) g\circ f=2x^4+4x^2+5 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{c}\right) f\circ f=x^4+2x^2+2and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{d}\right) g\circ g=8x^4+24x^2+21 and domain is \left(-\infty ,\infty \right) \end{gathered}$$

Part (a) Step 1. Given Information.

We have given,

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Part (a) Step 2. Concept.

A function which is depends on any other function we can call it as composite function.

$f\hspace{0.17em}\circ \hspace{0.17em}g=f\left(g\left(x\right)\right)$

Domain is the set of all input values where function is well defined and objective.

Part (a) Step 2. Explanation.

We have given,

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Using definition of the composite function,

$$\begin{gathered} f\hspace{0.17em}\circ \hspace{0.17em}g=f\left(g\left(x\right)\right) \\ =f\left(2x^2+3\right) \\ ={(2x^2+3)}^2+1 \\ =4x^4+12x^2+10 \end{gathered}$$

We have given functions f and g both are polynomial functions so the domain of both functions is set of real numbers.

Therefore domain of the composite function is also the set of real numbers.

Domain: $\left(-\infty ,\infty \right)$

Part (a) Step 4. Conclusion.

Hence, composite function of the functions $f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$ is $f\hspace{0.17em}\circ \hspace{0.17em}g=4x^4+12x^2+10$ and domain is $\left(-\infty ,\infty \right)$.

Part (b) Step 1. Explanation.

We have given,

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Using definition of the composite function,

$$\begin{gathered} g\hspace{0.17em}\circ \hspace{0.17em}f=g\left(f\left(x\right)\right) \\ =g\left(x^2+1\right) \\ =2{\left(x^2+1\right)}^2+3 \\ =2x^4+4x^2+5 \end{gathered}$$

We have given functions f and g both are polynomial functions so the domain of both functions is set of real numbers.

Therefore domain of the composite function is also the set of real numbers.

Domain: $\left(-\infty ,\infty \right)$

Part (b) Step 2. Conclusion.

Hence, composition of the functions $f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$ is $g\hspace{0.17em}\circ \hspace{0.17em}f=2x^4+4x^2+5$and domain is $\left(-\infty ,\infty \right).$

Part (c) Step 1. Explanation.

We have given,

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Using definition of the composite function,

$$\begin{gathered} f\hspace{0.17em}\circ \hspace{0.17em}f=f\left(f\left(x\right)\right) \\ =f\left(x^2+1\right) \\ ={\left(x^2+1\right)}^2+1 \\ =x^4+2x^2+2 \end{gathered}$$

We have given functions f and g both are polynomial functions so the domain of both functions is set of real numbers.

Therefore domain of the composite function is also the set of real numbers.

Domain: $\left(-\infty ,\infty \right)$

Part (c) Step 2. Conclusion.

Hence, composite function of the function $f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$ is $f\hspace{0.17em}\circ \hspace{0.17em}f=x^4+2x^2+2$ and domain is $\left(-\infty ,\infty \right).$

Part (d) Step 1. Explanation.

We have given,

$f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$

Using definition of the composite function,

$$\begin{gathered} g\hspace{0.17em}\circ \hspace{0.17em}g=g\left(g\left(x\right)\right) \\ =g\left(2x^2+3\right) \\ =2{\left(2x^2+3\right)}^2+3 \\ =8x^4+24x^2+21 \end{gathered}$$

We have given functions f and g both are polynomial functions so the domain of both functions is set of real numbers.

Therefore domain of the composite function is also the set of real numbers.

Domain: $\left(-\infty ,\infty \right)$

Part (d) Step 2. Conclusion.

Hence, composite function of the function $f\left(x\right)=x^2+1,\hspace{0.17em}g\left(x\right)=2x^2+3$ is $g\hspace{0.17em}\circ \hspace{0.17em}g=8x^4+24x^2+21$ and domain of the function is $\left(-\infty ,\infty \right).$