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Suppose that the functions
f and
g are defined as follows.

{:[f(x)=(9)/(2x)","x!=0],[g(x)=5x-4]:}
Find the compositions
f@f and
g@g.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all
x in the dor

{:[(f@f)(x)=◻],[(g@g)(x)=◻]:}

Suppose that the functions $f$ and $g$ are defined as follows. $\newline$ $\begin{array}{l} f(x)=\frac{9}{2 x}, x \neq 0 \\ g(x)=5 x-4 \end{array}$ $\newline$ Find the compositions $f \circ f$ and $g \circ g$ . $\newline$ Simplify your answers as much as possible. $\newline$ (Assume that your expressions are defined for all $x$ in the dor $\newline$ $\begin{array}{l} (f \circ f)(x)=\square \\ (g \circ g)(x)=\square \end{array}$

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Compare linear and exponential growth

Full solution

Q. Suppose that the functions

f

and

g

are defined as follows.

\newline

\begin{array}{l} f(x)=\frac{9}{2 x}, x \neq 0 \\ g(x)=5 x-4 \end{array}

\newline

Find the compositions

f \circ f

and

g \circ g

.

\newline

Simplify your answers as much as possible.

\newline

(Assume that your expressions are defined for all

x

in the dor

\newline

\begin{array}{l} (f \circ f)(x)=\square \\ (g \circ g)(x)=\square \end{array}

Calculate f@f(x): Calculate f@f(x), which means f(f(x)). $\newline$ First, substitute f(x) into f: $\newline$ $f(x) = \frac{9}{2x}$ , $\newline$ $f(f(x)) = f\left(\frac{9}{2x}\right)$ . $\newline$ Now, substitute $\frac{9}{2x}$ for x in f(x): $\newline$ $f\left(\frac{9}{2x}\right) = \frac{9}{2*\left(\frac{9}{2x}\right)}$ . $\newline$ Simplify the expression: $\newline$ $= \frac{9}{\frac{18}{2x}}$ $\newline$ $= 9 * \left(\frac{2x}{18}\right)$ $\newline$ $= \frac{9 * 2x}{18}$ $\newline$ $= \frac{x}{1}$ $\newline$ $= x$ .
Calculate $g@g(x)$ : Calculate $g@g(x)$ , which means $g(g(x))$ . $\newline$ First, substitute $g(x)$ into $g$ : $\newline$ $g(x) = 5x - 4$ , $\newline$ $g(g(x)) = g(5x - 4)$ . $\newline$ Now, substitute $5x - 4$ for $x$ in $g(x)$ : $\newline$ $g@g(x)$ $0$ $\newline$ $g@g(x)$ $1$ $\newline$ $g@g(x)$ $2$ .

More problems from Compare linear and exponential growth

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Jill bought a used motorcycle from a seller online for $1,200. The seller will charge her $5 a day to store the motorcycle at his house until she is able to pick it up. You can use a function to describe the total amount of money Jill will owe the seller if she waits

x

days to pick up the motorcycle.

\newline

Write an equation for the function. If it is linear, write it in the form

g(x) = mx + b

. If it is exponential, write it in the form

g(x) = a(b)^x

\newline

g(x) = \underline{\hspace{3em}}

Posted 5 months ago

Question

f(t)=-2t-7

change over the interval from

t=-7

t=-6

\newline

\newline

\left[\text{f(t) decreases by } 2\right]

\newline

\left[\text{f(t) increases by } 2\right]

\newline

\left[\text{f(t) increases by } 200\%\right]

\newline

\left[\text{f(t) decreases by a factor of } 2\right]

Posted 5 months ago

Question

g(t)=9^t

change over the interval from

t=1

t=3

\newline

\newline

\left[\text{g(t) decreases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by a factor of } 81\right]

\newline

\left[\text{g(t) increases by } 18\%\right]

\newline

\left[\text{g(t) decreases by } 9\%\right]

Posted 5 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A)f(x) = 8x + 3.3

\newline

(B)g(x) = 3.3^x - 5

Posted 6 months ago

Question

Both of these functions grow as

x

gets larger and larger. Which function eventually exceeds the other?

\newline

\newline

(A) \ f(x) = 2x + 9

\newline

(B)\ g(x) = 2^x

Posted 6 months ago

Question

f(x)

g(x)

are differentiable.

\newline

h(x)

h(x)=\frac{g(x)}{f(x)}

\newline

f(8)=1

f'(8)=-6

g(8)=8

g'(8)=-10

h'(8)

\newline

Simplify any fractions.

\newline

h'(8)=

Posted 6 months ago

Question

p(x)

q(x)

are differentiable.

\newline

r(x)

r(x)= \frac{p(x)}{q(x)}

\newline

p(3)= 2

p'(3)= 4

q(3)= 6

q'(3)= 9

r'(3)

\newline

Simplify any fractions.

\newline

r'(3)=

Posted 7 months ago

Question

u(x)

v(x)

are differentiable.

\newline

w(x)

w(x)= \frac{u(x)}{v(x)}

\newline

u(5)= 3

u'(5)= -2

v(5)= 7

v'(5)= 1

w'(5)

\newline

Simplify any fractions.

\newline

w'(5)=

Posted 7 months ago

Question

a(x)

b(x)

are differentiable.

\newline

c(x)

c(x)= \frac{a(x)}{b(x)}

\newline

a(2)= 4

a'(2)= 5

b(2)= 1

b'(2)= -3

c'(2)

\newline

Simplify any fractions.

\newline

c'(2)=

Posted 7 months ago

Question

m(x)

n(x)

are differentiable.

\newline

o(x)

o(x)= \frac{m(x)}{n(x)}

\newline

m(7)= 2

m'(7)= -1

n(7)= 4

n'(7)= 8

o'(7)

\newline

Simplify any fractions.

\newline

o'(7)=

Posted 7 months ago

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