Question

find the domain of the logarithmic function. (enter your answer using interval notation.) h(x)=ln(x + 3) find the x - intercept. (x,y)=( ) find the vertical asymptote. x = sketch the graph of the logarithmic function.

Explanation:

Step1: Find the domain

For the function $h(x)=\ln(x + 3)$, the argument of the logarithm must be positive. So we set $x+3>0$, which gives $x>- 3$. In interval - notation, the domain is $(-3,\infty)$.

Step2: Find the vertical asymptote

The vertical asymptote of $y = \ln(x + 3)$ occurs when the argument of the logarithm approaches 0. Setting $x+3 = 0$, we get $x=-3$. So the vertical - asymptote is $x=-3$.

Step3: Find the x - intercept

Set $h(x)=0$. Then $\ln(x + 3)=0$. Since $\ln1 = 0$, we have $x+3 = 1$, which gives $x=-2$. So the x - intercept is the point $(-2,0)$.

Step4: Sketch the graph

The graph of $y = \ln(x)$ is shifted 3 units to the left. It has a vertical asymptote at $x=-3$, passes through the point $(-2,0)$, and is an increasing function for $x>-3$.

Answer:

Domain: $(-3,\infty)$
Vertical asymptote: $x=-3$
x - intercept: $(-2,0)$