Problem

$61-72$ Graphing Logarithmic Functions Graph the …

Problem 64 Easy Difficulty

$61-72$ Graphing Logarithmic Functions Graph the function,
not by plotting points, but by starting from the graphs in Figures
4 and $9 .$ State the domain, range, and asymptote.
$$g(x)=\ln (x+2)$$

Answer

Vertical asymptote: $x=-2$
Domain: $D_{g}=(-2,+\infty)$
Range: $R_{g}=(-\infty,+\infty)$

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Algebra

Algebra and Trigonometry

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

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Video Transcript

All right. So we have been asked to plot the function. Why equals natural log of absolute value of X? In addition, we're supposed to find its domain, its range, and any aspirin totes that it may have. So let's calculate the domain first before we start plotting the domain of a log rhythmic function like this one is all things, uh, ex that make the function inside the log a rhythm strictly greater than zero. So for us, that's gonna be all the ex that make absolute value of X greater than zero. Now, absolute value of X is greater than zero a cz long as X is greater than zero ah, in magnitude. So the scent of X that satisfy this are all the X that are not equal to zero. So this is all the X that are not equal to zero. Those are the things that it's okay to put into the function, and that is its that its domain. So now for plotting this function, we're supposed to refer to figures four and nine in the book. So let's draw some coordinate axes here. Ah, and note a couple of things when X is greater than zero. So in this portion of the plot over here, the absolute value signs have no effect. Our function is just l and of X. There's no negative sign to go away under the absolute value. So it's just Ellen of X, And that means on this portion of the plot, we can refer to figure nine, which is a plot of Elena Vex. Ah, and so on this portion of the plot, our function looks like that where this is the X value one, this part looks like Ellen of X, and when X is less than zero, the absolute value does take effect. Our function looks like absolute value of whatever you put into it. So for each X value that's negative. We're going to end up with the same value. Ah, on the positive side for the number of equal magnitude. So this number over here would correspond to whatever value to function takes over here when we apply the absolute value. So let's let's see what our curve looks like. It's still going to go through, uh, negative one over here. It's just gonna look like this curve mirrored across the Y axis. So it's gonna come in like that and head down towards negative infinity as we approached the X axis. All right, as far as the Assam toads go, there's only one thing that we can't plug in here. There's only one element of the domain that has been excluded. X equals zero, and you can see in the plot that there's a vertical line that is not included in the plot here. So we have one ass and tote a vertical. Ask himto X equals zero in the range of this function eyes all the possible. Why values that could come out, Uh, to the right of the X axis. Here we are increasing. We increase up to positive infinity without bound and we take all the values tending down towards negative infinity. So the range of this function is all real numbers.

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