Question

Question 2 (2 points)Determine the following of the function y = 0.7log3(x).• DomainRange• X-intercept• Y-interceptAsymptote..

Answer 1

Recall that

$\log _ax$

is defined for all positive real numbers, therefore:

$\text{Dom}(0.7\log _3(x))=(0,\infty)\text{.}$

Also, since the range of log_3(x) is all real numbers, then:

$Ran(0.7\log _3(x))=(-\infty,\infty).$

Now, to find the x-intercept, we set y(x)=0:

$0.7\log _3(x)=0.$

Dividing the above equation by 0.7 we get:

$\begin{gathered} \frac{0.7}{0.7}\log _3(x)=\frac{0}{0.7}, \\ \log _3(x)=0. \end{gathered}$

Solving the above equation for x we get:

$\begin{gathered} 3^{\log _3(x)}=3^0, \\ x=1. \end{gathered}$

Therefore, the x-intercept has coordinates (1,0).

Since the function is only defined at (0,∞), there is no y-intercept.

Finally, the function has an asymptote at x=0.

Answer:

Domain:

$(0,\infty).$

Range:

$(-\infty,\infty).$

X-intercept:

$(1,0)\text{.}$

Y-intercept: There is no y-intercept.

Asymptote:

$x=0.$