Problem

$79-84$ Graphing Logarithmic Functions Draw the g…

Problem 79 Easy Difficulty

$79-84$ Graphing Logarithmic Functions Draw the graph of
the function in a suitable viewing rectangle, and use it to find the
domain, the asymptotes, and the local maximum and minimum
values.
$$y=\log _{10}\left(1-x^{2}\right)$$

Answer

$\begin{array}{ll}{\text { Domain }(-1,1)} & {\text { Range }(-\infty, 0)} \\ {\text { Asymptote } x=-1, x=1} & {\text { Local Max. }(0,0)}\end{array}$

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Algebra

Algebra and Trigonometry

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

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Video Transcript

so remember that the inside of a log rhythm function. So let's say, for example, we have we have long base x will need be or log based 10 of X. Well, the inside of this luxury function of this X has to be greater than or equal to or rather greater than zero greater than zero. And so we want we want this inside one minus X where to be greater than zero. So if we saw for this we have one is one is less than X squared. So one is greater than great are plus reminds one is is greater than greater than X. And so when we have this Well, what we mean is it is that X is is greater than negative one and X is less than positive. So we sketch this what we know we know if if if we convert this here into an exponential formal we would get If we convert the given blogger with Equation two exponential form. What we get is 10 to the so this would be 10 to me Why is equal to one minus X squared? So let's plug in zero and zero Well, 10 to 0 is one and we have one minus one minus zero square. So this is one is equal to one. So we know we have a point at the origin. This is accidents, is why And then we know that we're gonna have to Assen totes at negative one and positive one. So let's say this is, for example, negative one and or this is rather positive one. This is positive one, and this is negative. So we're going to be in the in the third and fourth quadrant. So our graph looks something like this is our asking toads. These are our ass. And toast on our graph looks something like like this. So now we confined our domain. You're which is from any number from negative 1 to 1. Our ranges from from negative infinity to zero. And our ascent oats are at X is equal to negative one and X is equal to one. And so these these are our solutions. And this is our sketch of the graph. And so remember we notice here that if we if we saw for the given log rhythmic equation right here, well, then we have Well, we would get is is X squared. X squared is equal to one minus tend to the wife. And if we take the square root But we have X is equal to plus are plus or minus plus or minus one minus tend to the wise. So we know that why cannot be greater than why cannot be greater than zero. So has the lesson. Zero and X can be both positive and negative. And so this is why we're in the third and fourth quadrant and this is how we get our solutions.

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