Question

hey,
can someone help me with this question?
sketch the graph of the function that satisfies the following conditions.
a) the domain is [-3,infinity)
b) range is (-1,10]
c) the curve passes through the point (-3,5)
d) the curve is smooth and continious

Answer 1

i dont understand how to make a curve in this question.

Answer 2

Domain and range mean values of x and y , respectively and you are given the values to consider in the graph.

Answer 3

\[\large f(x) ~~=~~ 11\dfrac{\sqrt{x+3}}{\sqrt{x+k}}-1\]

that satisfies \(a,b,d\) conditions
we can find the value of \(k\) by using the given point i guess..

Answer 4

may be not, as plugging in x=-3 makes that entire fraction 0

Answer 5

Anyway how did you come up with that equation?

Answer 6

domain is restricted to [-3, infty) and the max value has to be 10 so
\[f(x) = 10\dfrac{\sqrt{x+3}}{\sqrt{x+3}}\]

satisfies those two requirements

Answer 7

i can do it more easily with a piecewise defined function

Answer 8

i understand that, its just the function. i'm not sure about.

Answer 9

i can do it with a piece wise function as well. how did you come up with the function?

Answer 10

im thinking a bell curve (gaussian) function can do the job, in such a manner that it is not defined for x <-3

Answer 11

y = 11 exp(-(x+2)^2/2) - 1

Answer 12

we can multiply that by sqrt(x+3)^2 / (x+3)

Answer 13

$$
\Large {

y = \frac{(\sqrt{x+3}~)~^2}{x+3}\cdot 11 e^{-(x+k)^2/2} - 1



}

$$

find k such that y(-3) = 5

Answer 14

\[y = 5.5\sin(\sqrt{x+3})+4.5\]

Answer 15

$$
\LARGE {

y = \frac{(\sqrt{x+3}~)~^2}{x+3}\cdot 11
e^{- \frac{ \left( ~x+ 3 - \sqrt{-2\ln\frac{6}{11} } ~ \right) ^2 } {2} } - 1



}

$$

Answer 16

@rational your range is [-1,10] (i think)

Answer 17

the range of sin(sqrt(x+3) is [-1,1]
and
range 5.5 * [-1,1] + 4.5 = [-1,10]


The function I posted suffers from a different drawback, it is not defined at x = -3
but lim x->-3 is 5

Answer 18

@E_S_J_F did you find a function that works, piecewise?

Answer 19

I think this satisfies all your conditions.
$$
\Large {
y = \begin{cases}
{
\frac{(\sqrt{x+3}~)~^2}{x+3}\cdot 11
e^{- \frac{ \left( ~x+ 3 - \sqrt{-2\ln\frac{6}{11} } ~ \right) ^2 } {2} } - 1 ~~\text{if x >-3}
\\~\\ ~~~~~~~~~~~~~~~~~~5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{if x = -3}


}

\end{cases}


}
$$

Answer 20

THe directions say to graph the function that satisfies the condition, so you don't actually have to find an explicit function.

Answer 21

Ahh we need to have an asymptote at bottom

Answer 22

right

Answer 23

also note that my function is continuous, and differentiable

Answer 24

but it would be nice to find a non-piecewise defined function that satisfies all the requirements , explicitly given

Answer 25

is there a way to stop a function that is normally defined on R in its domain, but not to exclude that value.
The trick i used was to multiply by sqrt(x+3)^2 /(x+3).
something similar