Question

find features such as x and y intercepts to sketch this function (using e or in is not allowed)

Answer 1

Let's find the x-intercept of the given function. This ocurrs when y is equal to zero, so we have

$0=-log_2(3(x+2))+4$

which gives

$4-log_2(3(x+2))=0$

Now, we can rewrite the number 4 as follows

$4=log_22^4$

So, by substituting this result into the above equation, we have

$log_22^4-log_2(3(x+2))=0$

From the quotient rule of the logarithms, it can be written as

$log_2\frac{2^4}{3(x+2)}=0$

From the property

$log_b1=0$

we can conclude that

$\frac{2^4}{3(x+2)}=1$

or equivalently,

$\frac{16}{3(x+2)}=1$

so, we have

$3(x+2)=16$

which gives

$\begin{gathered} x+2=\frac{16}{3} \\ then \\ x=\frac{16}{3}+2=7.3333 \end{gathered}$

So, we have obtained that the x-intercept is the point (7.333, 0).

Similarly, the y-intercept ocurrs at x=0, which implies that

$y=-log_2(3(0+2))+4$

or equivalently,

$y=-log_2(-6)+4$

However, for a real base (2 in our case) the logarithm is undefined. This fact and since the logarithm has negative coefficient mean that the graph of the function has the form:

As we can corroborate with the followin graph: