(c) f(40) = f(45) this is the problem on function notation im lost i skipped that zoom class when she was talking about it

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Horizontal asymptote at  tells us that the original graph has been translated down.since  ,  are positive.  also we can say that the point  has a smaller  -coordinate than  .this means that the  transformed exponential function is increasing as it goes to the right. as it goes from  to , the value of the exponential function increases. it goes from to something bigger, . therefore, as  . the left hand would be the side that approaches the horizontal asymptote.so the answer is d

You need to figure out at which x-value does the graph has a tangent line slope of -1.remember, the derivative is the slope of the tangent line. so find the derivative function and set it to equal -1 to find the x-value that will have a tangent line slope of 0.differentiate with the chain rule.    set the derivative to equal -1 and solve for x.      take log base 10 of both sides. also recall that  .    so the x-coord of the point of tangency is .the y-coordinate of this point of tangency can be calculated using the original function:     using point-slope form, the equation of the tangent line is      the x-intercept of this tangent line is when the graph is at y=0.    this rounds off to  .choice c.

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