Logarithms functions graphing.

[Louise Johnson]

Here are two logarithmic function graphing questions. They just seem to easy for some reason which makes me suspicious. Also I am unsure about the notation I have used. The question asks to state the Domain, Range, values of intercepts, and equations of asymptotes that exist.
Thank you for your help
Louise

Question #1 \(\displaystyle f(x)= - \log _2 (x + 2) + 3\)

My answers:

\(\displaystyle domain\left\{ {x \in R|x \ge - 2} \right\}\)
Range All real numbers
X intecept =-1.87
Y intecept= 4
equation of Asymptote x=-2

Question #2 \(\displaystyle g(x) = 2e^x - 2\)

My answer:

Domain =all real numbers
\(\displaystyle range\left\{ {x \in R|y \le 5} \right\}\)
X intercept = 0
Y Intercept = 0
Equation of Asymptote y=-2

 

[mark07]

Question #1

\(\displaystyle f(x)= - \log _2 (x + 2) + 3\)

My answers:

\(\displaystyle domain\left\{ {x \in R|x \ge - 2} \right\}\) x=-2 is not in the domain, it should be x>-2.

Range All real numbers

X intercept =-1.87 Solving for f(x)=0 gives x=6.

Y intercept= 4 Setting x=0 gives y=2.

equation of vertical Asymptote x=-2

Question #2

\(\displaystyle g(x) = 2e^x - 2\)

My answer:

Domain =all real numbers

Range \(\displaystyle \left\{ {x \in R|y \le 5} \right\}\) \(\displaystyle e^x\) has range \(\displaystyle (0,\infty)\), so does \(\displaystyle 2e^x\). And finally \(\displaystyle 2e^x-2\) has range \(\displaystyle (-2,\infty)\).

X intercept = 0

Y Intercept = 0

Equation of horizontal Asymptote y=-2

 

[soroban]

Hello, Louise!

Are you familiar with "transformations" of graphs?
I'll baby-step through the first one . . .

State the Domain, Range, values of intercepts, and equations of asymptotes that exist.

\(\displaystyle 1)\;f(x) \;= \;-\log _2(x\,+\,2)\,+\,3\)

You may be expected to know the graph of: \(\displaystyle \,y\;=\;\log_2(x)\)

        |
        |                               *
        |                      *
        |                *
        |           *
      --+-------*--------------------------
        |    *  1
        |  *
        | *
        |
        |*
        |

Domain: \(\displaystyle \,(0,\,\infty)\)
Range: \(\displaystyle \,(-\infty,\,\infty)\)
Intercept: \(\displaystyle \,(1,\,0)\)
Asymptote: \(\displaystyle \,x\,=\,0\) (y-axis)


The "minus" in front of the functon reflects the curve over the x-axis: \(\displaystyle \:y \:=\:-\log_2(x)\)

        |*
        |
        | *
        |  *
        |    *  1
      --+-------*-------------------------
        |           *
        |                *
        |                      *
        |                               *

Domain: \(\displaystyle \,(0,\,\infty)\)
Range: \(\displaystyle \,(-\infty,\,\infty)\)
Intercept: \(\displaystyle \,(1,\,0)\)
Asymptote: \(\displaystyle \,x\,=\,0\) (y-axis)


The \(\displaystyle (x\,+\,2)\) moves the graph 2 units to the left: \(\displaystyle \:y\:=\:-\log_2(x\,+\,2)\).

        :*               |
        :                |
        : *              |
        :  *             |
        :    * -1        |
      --+-------*--------+------------------
       -2           *    |
        :              -1*
        :                |     *
        :                |               *

Domain: \(\displaystyle \,(-2,\,\infty)\)
Range: \(\displaystyle \,(-\infty,\,\infty)\)
Intercepts: \(\displaystyle \,(-1,\,0),\;(0,\,-1)\)
Asymptote: \(\displaystyle \,x\,=\,-2\)

The \(\displaystyle +\,3\) moves the graph 3 units up: \(\displaystyle \:y \:=\:-\log_2(x\,+\,2)\,+\,3\)

        :*               |
        :                |
        : *              |
        :  *             |
        :    *           |
        :       *        |
        :           *    |2
        :                *
        :                |     *
        :                |              *
        :                |
      --+----------------+----------------------------------*-----
       -2                |                                  6

Domain: \(\displaystyle \,(-2,\,\infty)\)
Range: \(\displaystyle \,(-\infty,\,\infty)\)
Intercepts: \(\displaystyle \,(6,\,0),\0,\,2)\)
Asymptote: \(\displaystyle \,x\,=\,-2\)

 

[Louise Johnson]

Wow Thank you
I am gonna need some time to go over all this
Thank you
Louise