Questions: Sketch the graph of the following function, and state its domain and range. f(x) = ln(x+7) List ordered pairs that satisfy the equation. x -6 -4 -2 f(x) = ln(x+7) 0 1.1 1.6 (Round to the nearest tenth as needed.)

Solution Steps

The logarithmic function $f(x) = \ln(x+7)$ is defined only when the argument is positive. Therefore, the domain of $f(x)$ is determined by the inequality $x + 7 > 0$, which simplifies to $x > -7$. In interval notation, the domain is $(-7, \infty)$.

We are given the following $x$ values: -6, -4, and -2. We need to find the corresponding $f(x)$ values by substituting the given $x$ values into the equation $f(x) = \ln(x+7)$.

For $x=-6$: $f(-6) = \ln(-6+7) = \ln(1) = 0$ For $x=-4$: $f(-4) = \ln(-4+7) = \ln(3) \approx 1.1$ For $x=-2$: $f(-2) = \ln(-2+7) = \ln(5) \approx 1.6$

The ordered pairs are $(-6, 0)$, $(-4, 1.1)$, and $(-2, 1.6)$.

To graph the function $f(x) = \ln(x+7)$, we can use the points we found in the previous step: $(-6,0)$, $(-4,1.1)$, and $(-2, 1.6)$. We also know the function has a vertical asymptote at $x=-7$. Plot these points and sketch a smooth curve that approaches the vertical asymptote as $x$ approaches -7 from the right, and increases slowly as $x$ goes to infinity. The graph shown in the problem is consistent with this description.

Final Answer: