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Question Video: Identifying the Graph of a Logarithmic Function and Using It to Find an Approximate Value of the Function Mathematics • 10th Grade

Consider the function 𝑓(π‘₯) = logβ‚‚ (βˆ’π‘₯). Which graph represents this function? Which of the following is an approximated value of 𝑓(βˆ’0.2)? [A] 0.87 [B] 1.15 [C] 2.32 [D] βˆ’2.32 [E] undefined

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Video Transcript

Consider the function 𝑓 of π‘₯ equals log to the base two of negative π‘₯. Which graph represents this function?

There is also a second part to this question, which we’ll look at once we’ve completed the first. The function we’ve been asked to consider is a logarithmic function. We know that logarithmic graphs of the form 𝑦 equals log to the base 𝑛 of π‘₯, where 𝑛 is positive and not equal to one, all have the same general shape and certain key properties. That’s not exactly what we have here though because we have negative π‘₯ instead of π‘₯. However, we know that swapping π‘₯ for negative π‘₯, so multiplying the variable by negative one, corresponds to a reflection in the 𝑦-axis.

What we can do then is use the general properties of logarithmic graphs to sketch the graph of a second function, which we’ll call 𝑔 of π‘₯. And this will be equal to log to the base two of π‘₯. And then we’ll reflect this graph in the 𝑦-axis to find the graph of 𝑓 of π‘₯. So, let’s remind ourselves what those properties are. The graphs of 𝑦 equals log to the base 𝑛 of π‘₯, where 𝑛 is positive and not equal to one, all pass through the point one, zero. That is, the π‘₯-intercept of the graphs is one. And they each pass through the point with coordinates 𝑛, one. They all have the 𝑦-axis as a vertical asymptote. The functions are only defined for positive values of π‘₯, so their domain is the open interval from zero to ∞. But the functions can produce any real value, so their range is the open interval from negative ∞ to positive ∞.

Let’s use these properties then to sketch the graph of 𝑔 of π‘₯, which is log to the base two of π‘₯. And we’ll do this on coordinate grid (A). Firstly, the graph needs to pass through the point with coordinates one, zero. The general graphs pass through the points with coordinates 𝑛, one. So, in this case, that’s the point two, one. Joining these points and ensuring that the 𝑦-axis is a vertical asymptote gives the graph shown in green. Remember though that this is the graph of 𝑔 of π‘₯, which is log to the base two of π‘₯.

To find the graph of 𝑓 of π‘₯, log to the base two of negative π‘₯, we now need to reflect this graph in the 𝑦-axis. This graph will pass through the points negative one, zero and negative two, one and will also have the 𝑦-axis as a vertical asymptote. The graph of 𝑓 of π‘₯, therefore, looks like this. We can now compare this to the five options we were given. And when we do, we see that the correct graph for 𝑓 of π‘₯ is graph (C). In particular, it does pass through the points with coordinates negative one, zero and negative two, one.

Let’s now clear some space so we can answer the second part of the question. So we’ve identified the graph of 𝑓 of π‘₯.

And the second part of the question says β€œWhich of the following is an approximated value of 𝑓 of negative 0.2? (a) 0.87, (b) 1.15, (c) 2.32, (d) negative 2.32, or (e) undefined.”

The graph we’ve identified then is the graph of 𝑦 equals 𝑓 of π‘₯. So, we need to use it to find an approximation of 𝑓 of negative 0.2. That is, we need to work out the value of the function when π‘₯ is equal to negative 0.2. Looking carefully at the scale on the π‘₯-axis of the graph, we can see that there are five small squares between each integer value. And so, each small square represents one-fifth or 0.2. An π‘₯-value of negative 0.2 then is one small square to the left of the 𝑦-axis; that’s here.

To find the approximate value of 𝑓 of negative 0.2, we need to read down from this value to the graph and then read across to the 𝑦-axis. When we do, we see that the 𝑦-value is between negative two and negative three. And in fact, it’s closer to negative two than to negative three. Of the five options given, the only one that meets these criteria is option (d). So, we can say that an approximated value of 𝑓 of negative 0.2 is negative 2.32.

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