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The (natural) domain of a function is the set of all real numbers to which the function assigns a value
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You need to impose the (existence) conditions under which each operation (such as division, square root, ...) or function (such as logarithms, ...), present in the analytic expression of the given function, can have a value (e.g. you can't divide by 0, you can't take the square root of a negative number, ...)
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Determine if the graph of the function is invariant under some type of geometric transformation
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The function is even if its graph is symmetric with respect to the ordinate axis, i.e. the value of the function is invariant under change of sign of the independent variable The function is odd if its graph is symmetric with respect to the origin of coordinate axis, i.e. the value of the function changes sign under change of sign of the independent variable
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The function is periodic, with period T, if its graph is invariant under horizontal translation by an amount equal to T, i.e. its value does not change when adding T to the independent variable
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Determine the regions of the cartesian plane wher the graph of the fuction is placed
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If 0 is an element of the domain then the graph intersect the axis of ordinates at the point whose ordinate is the value associated to 0 by the function
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The intersections of the graph with the axis of abscissas correspond to the zeroes of the function: to find them you need to solve an equation
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The sign of the function allows to know if the corresponding point of the graph is above or below the axis of abscissas. You can obtain the set of positivity and the set of negativity of the function (together with the set of zeroes) by solving a single inequality
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Analyse the behaviour of the function near the accumulation points of its domain where it is not continuous or undefined by means of limits
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The left and right limits are different real numbers
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At least one among the left and right limits is infinite or does not exist
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The left and the right limit are the same real number
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Analyse the behaviour of the graph at infinity to check if it approximates some stright line
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The graph of a function has a vertical asymptote if and only if the limit of the function when the independent variable tends to a real number is infinite
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The graph of the function has anhorizontal asymptote if the limit of the function when the independent variable tends to infinity is a real number
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For a slant asymptote to the graph of a function to exists three conditions need to be met
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Find the regions of the domain where the function is increasing or decreasing
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The (first) derivative of the function hosts information about the slope of the line tangent to the graph, hence the study of the monotonicity of the function is performed by solving a suitable inequality
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The function is increasing in every interval of its domain where the sign of its (first) derivative is positive
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The function is decreasing in every interval of its domain where the sign of its (first) derivative is negative
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Determine the position of estremals and of inflection points with horizontal tangent
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Zeroes of the (first) derivative are called critical (or stationary) points: they correspond to points of the graph where the tangent line is horizontal and generally, but not always, they signal a change of monotonicity (in which case they are also called extrema)
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A local maximum, which is one f the two inds of extremum, is a critical point for which there exist a left neighbourhood where the (first) derivative of the function is positive, and a right neighbourhood where the derivative is negative
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A local minimum, which is one f the two kinds of extremum, is a critical point for which there exist a left neighbourhood where the (first) derivative of the function is negative, and a right neighbourhood where the derivative is positive
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A horizontal point of inflection is a critical point which has a punctured neighbourhood where the (first) derivative has constant sign, i.e. it is either positive in that neighbourhood, or it is negative in that neighbourhood
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Understand the "branches" of th graph near points where the function is continuous but it is not derivable
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A corner point is a (not isolated) point of continuity of the function where the left and right limits of the (first) derivative are distinct real numbers
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A cusp is a (not isolated) point of continuity of the function where the left and right limits of the (first) derivative are infinite with different signs
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A vertical point of inflection is a (not isolated) point of continuity of the function where the left and right limits of the (first) derivative are infinite with the same sign
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Find in which way the graph curves with respect to the tangent line
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The second derivative of the function hosts information about the concavity of the graph, i.e. about which side of the tangent line the graph stays in a neighbourhood of the point. The function is concave up, or convex, at a point in a punctured neighbourhood of which the graph is above of the tangent line, whereas if the graph is below of the tangent line there the function is concave down, or concave. The study of the concavity of the function is performed by solving a suitable inequality
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The function is concave up (convex) in every interval of its domain where the sign of its second derivative is positive
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The function is concave down (concave) in every interval of its domain where the sign of its second derivative is negative
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Determine the type of the points whhere the graph changes its concavity type
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Zeroes of the second derivative signal a possible change of behaviour of the function with respect to concavity. If the sign of the second derivative changes crossing a zeroe then the corresponding point of the graph is called apoint of inflection (the graph "flexes" at that point)
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An ascending point of inflection is a point for which there exist a left neighbourhood where the second derivative of the function is negative, and a right neighbourhood where the second derivative is positive. Therefore, locally the function is concave to the left of a descending point of inflection, while it is convex to its right
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A descending point of inflection is a point for which there exist a left neighbourhood where the second derivative of the function is positive, and a right neighbourhood where the second derivative is negative. Therefore, locally the function is convex to the left of a descending point of inflection, while it is concave to its right
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All the information collected during the study of the function allows to sketch a probable graph representing geometrically the main properties of the function
