using the pictures provided solve For f(x) = 3x −9 3√x, find the domain, x-intercept(s), y-intercept, symmetry, vertical asymptotes, horizontal or slant asymptotes, intervals of increase or decrease, local maximum and minimum values, intervals of concavity, and inflection points. If any of the above characteristics are not present in the function, state NONE. Then, using these characteristics, sketch the curve. must show all steps. A. Domain: Find the domain D of f, which is the set of numbers for which f(x) isdefined.B. Intercepts: Find the r- and y-intercepts of the graph of f, if they exist. The y-intercept is (0, f(0)) and is where the graph intersects the y-axis. The r-intercepts arethe solutions to f (x) = 0 and are of the form (c, 0) where f(c) = 0. The r-interceptsmay be omitted if f(x) = 0 is difficult to solve.C. Symmetry:(i) If f(-x) = f(x) for all a in D, then f is an even function and its curve issymmetric about the y-axis. For the graph of an even function, the graph of thecurve for r 0 can be reflected over the y-axis to complete the graph. Commonexamples of graphs with y-axis symmetry are y = x?, y = |r], and y = cos r.(ii) If f(-x) = -f(x) for all z in D, then f is an odd function and its curveis symmetric about the origin. For the graph of an odd function, the graphof the curve for x 0 can be rotated 180° about the origin to complete thegraph. Common examples of graphs with origin symmetry are y = , y = x andy = sin r.(iii) If f(r +p) = f (r) for all r in D, where p 0 is a constant, then f is a periodicfunction and the smallest such number p satisfying this condition is the periodof f. The graph of the function over an interval of p can then be translated tocomplete the rest of the graph. Common examples of graphs that are periodicare the trigonometric graphs y = sin r and y = cos r.D. Asymptotes:(i) Horizontal Asymptotes: If either lim f(r) = L or lim f(x) = L, then y = Lis a horizontal asymptote of y = f (x). If either of these limits approaches too,there is not a horizontal asymptote but it is still useful information to know interms of sketching the graph of the function since it indicates how the sketch willstart and end. If f(a) is not defined and a is an endpoint of the domain of f, wecan find lim f(x) or lim f(x) instead. (ii) Vertical Asymptotes: The line r = a is a vertical asymptote of y = f(x) if at leastone of the four following statements is true:lim f(x) = 0lim f(r) = 00エ→a-lim f(x) = -0lim f(r) = -o0エ→a+For a rational function, a shortcut for finding its vertical asymptotes is to factorthe numerator and denominator. Then cancel any common factors (the r-valuewhere a common factor is zero typically results in a hole in the graph). Thenset the remaining factors in the denominator equal to zero and solve to get thevertical asymptotes.(iii) Slant Asymptotes: This is discussed later in the section.E. Intervals of Increase or Decrease: Find f'(x) and use the Increasing/DecreasingTest. The intervals on which f'(x) is positive, f is increasing, and the intervals onwhich f'(x) is negative, f is decreasing.F. Local Maximum and Minimum Values: To find the local maximum and minimumvalues of f, start by finding the critical numbers of f (numbers c such that f(c) existsand f'(c) = 0 or f'(c) does not exist). Then apply the First Derivative Test: If f'changes from positive to negative at critical number c, then f(c) is a local maximum. Iff' changes from negative to positive at critical number c, then f(c) is a local minimum.An alternative is to use the Second Derivative Test.G. Concavity and Inflection Points: To determine the concavity of f, find f"(x) andapply the Concavity Test. The graph of f is concave upward where f"(x) 0 andconcave downward where f"(x)

Question

using the pictures provided solve For f(x) = 3x −9 3√x, find the domain, x-intercept(s), y-intercept, symmetry, vertical asymptotes, horizontal or slant asymptotes, intervals of increase or decrease, local maximum and minimum values, intervals of concavity, and inflection points. If any of the above characteristics are not present in the function, state NONE. Then, using these characteristics, sketch the curve. must show all steps. A. Domain: Find the domain D of f, which is the set of numbers for which f(x) isdefined.B. Intercepts: Find the r- and y-intercepts of the graph of f, if they exist. The y-intercept is (0, f(0)) and is where the graph intersects the y-axis. The r-intercepts arethe solutions to f (x) = 0 and are of the form (c, 0) where f(c) = 0. The r-interceptsmay be omitted if f(x) = 0 is difficult to solve.C. Symmetry:(i) If f(-x) = f(x) for all a in D, then f is an even function and its curve issymmetric about the y-axis. For the graph of an even function, the graph of thecurve for r > 0 can be reflected over the y-axis to complete the graph. Commonexamples of graphs with y-axis symmetry are y = x?, y = |r], and y = cos r.(ii) If f(-x) = -f(x) for all z in D, then f is an odd function and its curveis symmetric about the origin. For the graph of an odd function, the graphof the curve for x > 0 can be rotated 180° about the origin to complete thegraph. Common examples of graphs with origin symmetry are y = , y = x andy = sin r.(iii) If f(r +p) = f (r) for all r in D, where p > 0 is a constant, then f is a periodicfunction and the smallest such number p satisfying this condition is the periodof f. The graph of the function over an interval of p can then be translated tocomplete the rest of the graph. Common examples of graphs that are periodicare the trigonometric graphs y = sin r and y = cos r.D. Asymptotes:(i) Horizontal Asymptotes: If either lim f(r) = L or lim f(x) = L, then y = Lis a horizontal asymptote of y = f (x). If either of these limits approaches too,there is not a horizontal asymptote but it is still useful information to know interms of sketching the graph of the function since it indicates how the sketch willstart and end. If f(a) is not defined and a is an endpoint of the domain of f, wecan find lim f(x) or lim f(x) instead. (ii) Vertical Asymptotes: The line r = a is a vertical asymptote of y = f(x) if at leastone of the four following statements is true:lim f(x) = 0lim f(r) = 00エ→a-lim f(x) = -0lim f(r) = -o0エ→a+For a rational function, a shortcut for finding its vertical asymptotes is to factorthe numerator and denominator. Then cancel any common factors (the r-valuewhere a common factor is zero typically results in a hole in the graph). Thenset the remaining factors in the denominator equal to zero and solve to get thevertical asymptotes.(iii) Slant Asymptotes: This is discussed later in the section.E. Intervals of Increase or Decrease: Find f'(x) and use the Increasing/DecreasingTest. The intervals on which f'(x) is positive, f is increasing, and the intervals onwhich f'(x) is negative, f is decreasing.F. Local Maximum and Minimum Values: To find the local maximum and minimumvalues of f, start by finding the critical numbers of f (numbers c such that f(c) existsand f'(c) = 0 or f'(c) does not exist). Then apply the First Derivative Test: If f'changes from positive to negative at critical number c, then f(c) is a local maximum. Iff' changes from negative to positive at critical number c, then f(c) is a local minimum.An alternative is to use the Second Derivative Test.G. Concavity and Inflection Points: To determine the concavity of f, find f&#34;(x) andapply the Concavity Test. The graph of f is concave upward where f&#34;(x) > 0 andconcave downward where f&#34;(x) < 0. Inflection points are points on the graph of fwhere the direction of concavity changes.H. Sketch the Curve: Use the information from A-G to sketch the graph by completingthe following steps:(i) Sketch any asymptotes as dashed lines.(ii) Plot any points such as the intercepts, local extrema, and inflection points.(iii) Pass a curve through these points, illustrating whether the graph rises or fallsbased on E and the appropriate concavity based on G. The graph should alsoapproach the asymptotes in the manner indicated by the limits.(iv) Additional points can be included in any areas that are missing features.

Accepted Answer

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There are many subparts in the given question so I will answer the first three of them and…