Q 30. - 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. 

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{Part} \left(\mathrm{a}\right) f\circ g=-2x+4 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{b}\right) g\circ f=-2x-3 and domain is \left(-\infty ,\infty \right) \\ \mathrm{Part} \left(\mathrm{c}\right) f\circ f=x and domain is \left(-\infty ,\infty \right) \\ Part \left(d\right) g\circ g=4x-12 and domain is \left(-\infty ,\infty \right) \end{gathered}$$ 

Part (a) Step 1. Given Information.

We have given,

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Part (a) Step 2. Concept.

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

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

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

Part (a) Step 3. Explanation.

We have given,

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Using definition of the composite function,

$$\begin{gathered} (f\circ g)\left(x\right)=f\left(g\right(x\left)\right) \\ =f(2x-4) \\ =-(2x-4) \\ =-2x+4 \end{gathered}$$

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 $f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$ is $\left(f\hspace{0.17em}\circ \hspace{0.17em}g\right)=-2x+4$ and domain is $\left(-\infty ,\infty \right)$.

Part (b) Step 1. Explanation

We have given,

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Using definition of the composite function,

$$\begin{gathered} \left(g\circ f\right)\left(x\right)=g\left(f\right(x\left)\right) \\ =g(-x) \\ =2(-x)-3 \\ =-2x-3 \end{gathered}$$

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

Part (c) Step 1. Explanation.

We have given,

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Using definition of the composite function,

$$\begin{gathered} (f\circ f)\left(x\right)=f\left(f\right(x\left)\right) \\ =f(-x) \\ =-(-x) \\ =x \end{gathered}$$

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 $f\left(x\right)=-x\hspace{0.17em}$with itself is $f\circ f=x$ and domain of the function is, $\left(-\infty ,\infty \right)$.

Part (d) Step 1. Explanation.

We have given,

$f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$

Using definition of the composite function,

$$\begin{gathered} (g\circ g)\left(x\right)=g\left(g\right(x\left)\right) \\ =g(2x-4) \\ =2(2x-4)-4 \\ =4x-8-4 \\ =4x-12 \end{gathered}$$

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 functions $f\left(x\right)=-x\hspace{0.17em}and\hspace{0.17em}g\left(x\right)=2x-4$ is

$g\circ g=4x-12$ and domain of the composite function is, $\left(-\infty ,\infty \right)$.