Q 32. - 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. 

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{Part} \left(\mathrm{a}\right) f\circ g=x^2+5 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{b}\right) g\circ f=x^2+2x+5 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{c}\right) f\circ f=x^2+4 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{d}\right) g\circ g=x^2+8x^2+20 and domain is \left(-\infty ,\infty \right) \end{gathered}$$ 

Part (a) Step 1. Given Information.

We have given,

$f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$

Part (a) Step 2. Concept used.

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 3. Explanation.

We have given, 

$f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$

Using definition of the composite function,

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

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: $\left(-\infty ,\hspace{0.17em}\infty \right)$

Part (a) Step 4. Conclusion.

Hence, composite function of the functions $f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$ is

$f\hspace{0.17em}\circ \hspace{0.17em}g =x^2+5$ and domain is $\left(-\infty ,\hspace{0.17em}\infty \right)$.

Part (b) Step 1. Explanation.

We have given, 

$f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$

Using definition of the composite function,

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

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: $\left(-\infty ,\hspace{0.17em}\infty \right)$

Part (b) Step 2. Conclusion.

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

Part (c) Step 1. Explanation.

We have given, 

$f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$

Using definition of the composite function,

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

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:  $\left(-\infty ,\hspace{0.17em}\infty \right)$

Part (c) Step 2. Conclusion.

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

Part (d) Step 1. Explanation.

We have given, 

$f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$

Using definition of the composite function,

$$\begin{gathered} g\hspace{0.17em}\circ \hspace{0.17em}g=g\left(g\left(x\right)\right) \\ = g\left(x^2+4\right) \\ ={\left(x^2+4\right)}^2+4 \\ =x^4+8x^2+20 \end{gathered}$$

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: $\left(-\infty ,\hspace{0.17em}\infty \right)$

Part (d) Step 2. Conclusion.

Hence, composite function of the function $f\left(x\right)=x+1,\hspace{0.17em}g\left(x\right)=x^2+4$ is $g\hspace{0.17em}\circ \hspace{0.17em}g=x^4+8x^2+20$ and domain is $\left(-\infty ,\hspace{0.17em}\infty \right)$.