Question

more help

Answer 1

its ok beautiful

Answer 2

c d a in that exact order

Answer 3

question #1
we can write this:
\[\Large g\left( {f\left( x \right)} \right) = \sqrt {f\left( x \right) + 4} = \sqrt {\left( {{x^2} + 3} \right) + 4} = \sqrt {{x^2} + 7} \]

Answer 4

so, we have:
\[\Large g\left( {f\left( 3 \right)} \right) = \sqrt {{3^2} + 7} = \sqrt {9 + 7} = ...?\]

Answer 5

hmm hold on

Answer 6

c d a

Answer 7

16

Answer 8

and what is \(\sqrt{16}=...?\)

Answer 9

23

Answer 10

I think that the square root of 16 is 4, am I right?

Answer 11

yes

Answer 12

so, what is the right option?

Answer 13

A

Answer 14

correct!

Answer 15

hey

Answer 16

okay what about 2

Answer 17

question #2
we can write this:
\[\large f\left( {g\left( x \right)} \right) = \sqrt {g\left( x \right) - 3} = \sqrt {4x - {x^2} - 3} = \sqrt {{x^2} + 4x - 3} \]

Answer 18

how do i solve this?

Answer 19

now, the domain of such composed function, is given by all values of \(x\), such that:
\[\huge {x^2} + 4x - 3 \geqslant 0\]
since the square root of negative numbers doesn't exist.
So we have to solve such inequality

Answer 20

sorry I have made an error, here is the right equation:
\[\large f\left( {g\left( x \right)} \right) = \sqrt {g\left( x \right) - 3} = \sqrt {4x - {x^2} - 3} = \sqrt { - \left( {{x^2} - 4x + 3} \right)} \]

Answer 21

so, we have to solve this inequality:
\[ - \left( {{x^2} - 4x + 3} \right) \geqslant 0\]
which is equivalent to this one:
\[\huge {x^2} - 4x + 3 \leqslant 0\]

Answer 22

oh okay

Answer 23

please try

Answer 24

i dont know how

Answer 25

hint:
please try to solve this equation:
\[\huge {x^2} - 4x + 3 = 0\]

Answer 26

x=1 and x=3

Answer 27

correct!
Now, from the general theory of inequality, the solution of:
\[\Large {x^2} - 4x + 3 \leqslant 0\]
is given by the subsequent interval:
\[\Large x \in \left( {1,\;3} \right)\]
namely:
\[\Large 1 \leqslant x \leqslant 3\]
geometrically speaking, we can visualize such solution, as below: