3.4 Composition of functions  (Page 7/9)

 Page 7 / 9

What is the composition of two functions, $\text{\hspace{0.17em}}f\circ g?$

If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.

Yes. Sample answer: Let Then $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=f\left(x-1\right)=\left(x-1\right)+1=x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=g\left(x+1\right)=\left(x+1\right)-1=x.\text{\hspace{0.17em}}$ So $\text{\hspace{0.17em}}f\circ g=g\circ f.$

How do you find the domain for the composition of two functions, $\text{\hspace{0.17em}}f\circ g?$

Algebraic

For the following exercises, determine the domain for each function in interval notation.

Given and find and

$\left(f+g\right)\left(x\right)=2x+6,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(f-g\right)\left(x\right)=2{x}^{2}+2x-6,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(fg\right)\left(x\right)=-{x}^{4}-2{x}^{3}+6{x}^{2}+12x,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{{x}^{2}+2x}{6-{x}^{2}},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,-\sqrt{6}\right)\cup \left(-\sqrt{6},\sqrt{6}\right)\cup \left(\sqrt{6},\infty \right)$

Given and find $\text{\hspace{0.17em}}f+g,\text{\hspace{0.17em}}f-g,\text{\hspace{0.17em}}fg,\text{\hspace{0.17em}}$ and

Given and find and

$\left(f+g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}+1}{2x},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(f-g\right)\left(x\right)=\frac{4{x}^{3}+8{x}^{2}-1}{2x},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(fg\right)\left(x\right)=x+2,\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=4{x}^{3}+8{x}^{2},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right)$

Given $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x-4}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{6-x},\text{\hspace{0.17em}}$ find and

Given $\text{\hspace{0.17em}}f\left(x\right)=3{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-5},\text{\hspace{0.17em}}$ find and

$\left(f+g\right)\left(x\right)=3{x}^{2}+\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(f-g\right)\left(x\right)=3{x}^{2}-\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(fg\right)\left(x\right)=3{x}^{2}\sqrt{x-5},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left[5,\infty \right)$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{3{x}^{2}}{\sqrt{x-5}},\text{\hspace{0.17em}}$ domain: $\text{\hspace{0.17em}}\left(5,\infty \right)$

Given $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=|x-3|,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}\frac{g}{f}.\text{\hspace{0.17em}}$

For the following exercise, find the indicated function given $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=3x-5.\text{\hspace{0.17em}}$

1. $f\left(g\left(2\right)\right)$
2. $f\left(g\left(x\right)\right)$
3. $g\left(f\left(x\right)\right)$
4. $\left(g\circ g\right)\left(x\right)$
5. $\left(f\circ f\right)\left(-2\right)$

a. 3; b. $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=2{\left(3x-5\right)}^{2}+1;\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)=6{x}^{2}-2;\text{\hspace{0.17em}}$ d. $\text{\hspace{0.17em}}\left(g\circ g\right)\left(x\right)=3\left(3x-5\right)-5=9x-20;\text{\hspace{0.17em}}$ e. $\text{\hspace{0.17em}}\left(f\circ f\right)\left(-2\right)=163$

For the following exercises, use each pair of functions to find $\text{\hspace{0.17em}}f\left(g\left(x\right)\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(f\left(x\right)\right).\text{\hspace{0.17em}}$ Simplify your answers.

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

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

$f\left(g\left(x\right)\right)=\sqrt{{x}^{2}+3}+2,\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=x+4\sqrt{x}+7$

$f\left(x\right)=|x|,\text{\hspace{0.17em}}g\left(x\right)=5x+1$

$f\left(x\right)=\sqrt[3]{x},\text{\hspace{0.17em}}g\left(x\right)=\frac{x+1}{{x}^{3}}$

$f\left(g\left(x\right)\right)=\sqrt[3]{\frac{x+1}{{x}^{3}}}=\frac{\sqrt[3]{x+1}}{x},\text{\hspace{0.17em}}g\left(f\left(x\right)\right)=\frac{\sqrt[3]{x}+1}{x}$

$f\left(x\right)=\frac{1}{x-6},\text{\hspace{0.17em}}g\left(x\right)=\frac{7}{x}+6$

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

For the following exercises, use each set of functions to find $\text{\hspace{0.17em}}f\left(g\left(h\left(x\right)\right)\right).\text{\hspace{0.17em}}$ Simplify your answers.

$f\left(x\right)={x}^{4}+6,\text{\hspace{0.17em}}$ $g\left(x\right)=x-6,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=\sqrt{x}$

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

$f\left(g\left(h\left(x\right)\right)\right)={\left(\frac{1}{x+3}\right)}^{2}+1$

Given $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=x-3,\text{\hspace{0.17em}}$ find the following:

1. $\left(f\circ g\right)\left(x\right)$
2. the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation
3. $\left(g\circ f\right)\left(x\right)$
4. the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)\text{\hspace{0.17em}}$
5. $\left(\frac{f}{g}\right)x$

Given $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{2-4x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=-\frac{3}{x},\text{\hspace{0.17em}}$ find the following:

1. $\left(g\circ f\right)\left(x\right)$
2. the domain of $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation

a. $\text{\hspace{0.17em}}\left(g\circ f\right)\left(x\right)=-\frac{3}{\sqrt{2-4x}};\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(-\infty ,\frac{1}{2}\right)$

Given the functions $\text{\hspace{0.17em}}f\left(x\right)=\frac{1-x}{x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{1+{x}^{2}},$ find the following:

1. $\left(g\circ f\right)\left(x\right)$
2. $\left(g\circ f\right)\left(\text{2}\right)$

Given functions $\text{\hspace{0.17em}}p\left(x\right)=\frac{1}{\sqrt{x}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}m\left(x\right)={x}^{2}-4,\text{\hspace{0.17em}}$ state the domain of each of the following functions using interval notation:

1. $\frac{p\left(x\right)}{m\left(x\right)}$
2. $p\left(m\left(x\right)\right)$
3. $m\left(p\left(x\right)\right)$

a. $\text{\hspace{0.17em}}\left(0,2\right)\cup \left(2,\infty \right);\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(-\infty ,-2\right)\cup \left(2,\infty \right);\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}\left(0,\infty \right)$

Given functions $\text{\hspace{0.17em}}q\left(x\right)=\frac{1}{\sqrt{x}}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)={x}^{2}-9,\text{\hspace{0.17em}}$ state the domain of each of the following functions using interval notation.

1. $\frac{q\left(x\right)}{h\left(x\right)}$
2. $q\left(h\left(x\right)\right)$
3. $h\left(q\left(x\right)\right)$

For $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\sqrt{x-1},\text{\hspace{0.17em}}$ write the domain of $\text{\hspace{0.17em}}\left(f\circ g\right)\left(x\right)\text{\hspace{0.17em}}$ in interval notation.

$\left(1,\infty \right)$

For the following exercises, find functions $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ so the given function can be expressed as $\text{\hspace{0.17em}}h\left(x\right)=f\left(g\left(x\right)\right).$

$h\left(x\right)={\left(x+2\right)}^{2}$

$h\left(x\right)={\left(x-5\right)}^{3}$

sample: $\begin{array}{l}f\left(x\right)={x}^{3}\\ g\left(x\right)=x-5\end{array}$

$h\left(x\right)=\frac{3}{x-5}$

$h\left(x\right)=\frac{4}{{\left(x+2\right)}^{2}}$

sample: $\begin{array}{l}f\left(x\right)=\frac{4}{x}\hfill \\ g\left(x\right)={\left(x+2\right)}^{2}\hfill \end{array}$

$h\left(x\right)=4+\sqrt[3]{x}$

$h\left(x\right)=\sqrt[3]{\frac{1}{2x-3}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[3]{x}\\ g\left(x\right)=\frac{1}{2x-3}\end{array}$

$h\left(x\right)=\frac{1}{{\left(3{x}^{2}-4\right)}^{-3}}$

$h\left(x\right)=\sqrt[4]{\frac{3x-2}{x+5}}$

sample: $\begin{array}{l}f\left(x\right)=\sqrt[4]{x}\\ g\left(x\right)=\frac{3x-2}{x+5}\end{array}$

what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
I want to know partial fraction Decomposition.
classes of function in mathematics
divide y2_8y2+5y2/y2
wish i knew calculus to understand what's going on 🙂
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE