Question 116021
b) 


<b>Description of transformation</b>:


Remember, {{{f(x)}}} is the same as y. So this means {{{y=log(x)}}}



Now if we negate both sides to get  {{{-y=-log(x)}}}


So {{{g(x)}}} is simply making each y coordinate becomes it's opposite. So something like (0,2) becomes (0,-2) and (3,-2) becomes (3,2), etc



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Answer:

So what's happening is that the graph is being reflected over the x-axis



Notice if we graph {{{f(x)}}} and {{{g(x)}}}, we get


{{{ graph( 500, 500, -10, 10, -10, 10, log(10,(x)), -log(10,(x)))}}}  Graph of {{{f(x)=log(10,(x))}}} (red) and  {{{g(x)=-log(10,(x))}}} (green)


and we can visually verify the transformation



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<b>Vertical Asymptote</b>:


From the graph, we can see that the vertical asymptote is {{{x=0}}}. Since the transformation reflected the graph across the x-axis, the vertical asymptote of {{{g(x)}}} is the same as the vertical asymptote of {{{f(x)}}} 


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Answer:


So the vertical asymptote of {{{g(x)}}} is {{{x=0}}}



We can verify this by looking at the graph above


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<b>x-intercept in (x, y) form</b>:


From the graph, we can see that the x-intercept of {{{f(x)}}} is (1,0). Since we've reflected everything with respect to the x-axis, the point on the x-axis is not affected. In other words the x-intercept of {{{g(x)}}} is the same as the x-intercept of {{{f(x)}}} 


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Answer:

So the x-intercept of {{{g(x)}}} is (1,0)



Once again, we can visually verify this if we look at the graph above