Let f(x) = 22-x. Fill answers in the blank. (1) The domain of f is (2)...
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the problem of finding the properties of the function fx 2x 1 The domain of f The domain of f is all nonnegative real numbers because the square root of a number is only defined for nonnegative numbers So the domain can be written as x 0 2 The intercepts yintercept The yintercept of a function is the point where the graph of the function crosses the yaxis This happens when x 0 In this case f0 20 0 Therefore the yintercept is at 0 0 xintercept The xintercept of a function is the point where the graph of the function crosses the xaxis This happens when fx 0 In this case there is no real number x for which 2x 0 Therefore the function fx 2x does not have an xintercept 3 Horizontal and vertical asymptotes Horizontal asymptote A horizontal asymptote is a line that the graph of the function approaches as x approaches positive or negative infinity In this case as x approaches infinity the term 2x becomes increasingly smaller compared to x Therefore the graph of fx approaches the yaxis as x approaches infinity So the horizontal asymptote is y 0 Vertical asymptote A vertical asymptote is a line that the graph of the function cannot cross but it gets arbitrarily close to the line as x approaches a certain value In this case there is no vertical asymptote for fx 2x because the function is defined for all nonnegative real numbers 4 Derivatives fx The derivative of fx can be found using the power rule of differentiation The power rule states that the derivative of xn is n xn1 In this case n 12 Therefore fx 12 x12 12x fx The second derivative of fx can be found by differentiating fx In this case fx 14x32 5 Critical numbers Critical numbers are the points where the derivative of the function is equal to zero or undefined In this case fx 0 for x 0 However fx is undefined at x 0 Therefore the only critical number of fx is x 0 6 Increasingdecreasing intervals Increasing A function is increasing on an interval where its derivative is positive In this case fx 0 for all x 0 Therefore fx is increasing on the interval 0 Decreasing A function is decreasing on an interval where its derivative is negative In this case fx does not exist at x 0 However fx 0 for all x in the interval 0 001 Therefore we can say that fx is decreasing on the interval 0 001 7 Local extrema Local maximum A local maximum is a point where the function changes from increasing to decreasing In this case fx is increasing on the interval 0 so it does not have a local maximum Local minimum A local minimum is a point where the function changes from decreasing to increasing In this case fx is decreasing on the interval 0 001 so it has a local minimum at the point 001 f001 8 Concavity Concave upward A function is concave upward on an interval where its second derivative is positive In this case fx 0 for all x 0 Therefore fx is concave upward on the interval 0 Concave downward A function is concave downward on an interval where its second derivative is negative fx never takes a negative value Therefore fx is never concave downward 9 Inflection points An inflection point is a point where the concavity of the function changes Since fx is always more detailed explanation of the properties of the function fx 2x Heres a breakdown of each point with additional insights 1 Domain Explanation The square root operation is only defined for nonnegative numbers because taking the square root of a negative number ... View the full answer
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