Question

In each of the following functions determine it's

A.domain

B.range

C.x-intercept

D.zeroes

E.vertical asymptote

1y=log5(x-1)+3

2.y=log(3x-2)

Answer 1

It is Assumed the logarithm given are to base 10.

$y=log5(x-1)+3$

it's Domain

For the log to be positive $x>1$

$(1,\infin)$

Range

All real numbers

$(-\infin,\infin)$

x-intercept

log $5(x-1)+3=0$

$log5(x-1)=-3$

$5x-5=0.001$

$x=1.0002$

$(1.0002,0)$

Zeros

At $y=0$

$x=1.0002$

Vertical Asymptote

For $5(x-1)>0$

$x>1$

Asymptote is $x=1$

$y=log(3x-2)$

Domain

For log to be positive $3x-2>0\implies\>x>\frac{2}{3}$

$(\frac{2}{3},\infin)$

Range

All real values

$(-\infin,\infin)$

x-intercept

log $(3x-2)=0$

$3x-2=1$

$x=1$

$(1,0)$

Zeros

For $y=0,\quad x=1$

Vertical Asymptote

$3x-2>0$

Vertical Asymptote is $x=\frac{2}{3}$