Study of a function

By a.zampa
  • Period: 100 to 199

    Determining the domain

    The (natural) domain of a function is the set of all real numbers to which the function assigns a value
  • 150

    Solve inequalities

    Solve inequalities
    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, ...)
  • Period: 200 to 299

    Determining symmetries

    Determine if the graph of the function is invariant under some type of geometric transformation
  • 233

    Parity (axial / central symmetry)

    Parity (axial / central symmetry)
    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
  • 267

    Periodicity (translation)

    Periodicity (translation)
    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
  • Period: 300 to 399

    Determining sign and intersections of the graph with coordinate axes

    Determine the regions of the cartesian plane wher the graph of the fuction is placed
  • 325

    Intersection with the axis of ordinates

    Intersection with the axis of ordinates
    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
  • 350

    Intersections with the axis of abscissas

    Intersections with the axis of abscissas
    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
  • 375

    Sign of the function

    Sign of the function
    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
  • Period: 400 to 499

    Classifying discontinuities/singularities

    Analyse the behaviour of the function near the accumulation points of its domain where it is not continuous or undefined by means of limits
  • 425

    Jump discontinuty / singularity (of the first kind)

    Jump discontinuty / singularity (of the first kind)
    The left and right limits are different real numbers
  • 450

    Essential discontiunity / singularity (of the second kind)

    Essential discontiunity / singularity (of the second kind)
    At least one among the left and right limits is infinite or does not exist
  • 475

    Removable discontinuity / singularity (of the third kind)

    Removable discontinuity / singularity (of the third kind)
    The left and the right limit are the same real number
  • Period: 500 to 599

    Determining asymptotes

    Analyse the behaviour of the graph at infinity to check if it approximates some stright line
  • 525

    Vertical asymptote

    Vertical asymptote
    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
  • 550

    Classifying inflection pointsHorizontal asymptote

    Classifying inflection pointsHorizontal asymptote
    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
  • 575

    Slant asymptote

    Slant asymptote
    For a slant asymptote to the graph of a function to exists three conditions need to be met
  • Period: 600 to 699

    Determining growth (monotonicity)

    Find the regions of the domain where the function is increasing or decreasing
  • 625

    Calculate the (first) derivative

    Calculate the (first) derivative
    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
  • 650

    Increasing function

    Increasing function
    The function is increasing in every interval of its domain where the sign of its (first) derivative is positive
  • 675

    Decreasing function

    Decreasing function
    The function is decreasing in every interval of its domain where the sign of its (first) derivative is negative
  • Period: 700 to 799

    Classifying stationary (critical) points

    Determine the position of estremals and of inflection points with horizontal tangent
  • 720

    Critical points

    Critical points
    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)
  • 740

    Local maximum

    Local maximum
    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
  • 760

    Local minimum

    Local minimum
    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
  • 780

    Horizontal point of inflection

    Horizontal point of inflection
    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
  • Period: 800 to 899

    Classifying points of non-derivability

    Understand the "branches" of th graph near points where the function is continuous but it is not derivable
  • 825

    Corner point

    Corner point
    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
  • 850

    Cusp

    Cusp
    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
  • 875

    Vertical point of inflection

    Vertical point of inflection
    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
  • Period: 900 to 999

    Determining the concavity

    Find in which way the graph curves with respect to the tangent line
  • 925

    Calculate the second derivative

    Calculate the second derivative
    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
  • 950

    Concave up (convex) function

    Concave up (convex) function
    The function is concave up (convex) in every interval of its domain where the sign of its second derivative is positive
  • 975

    Concave down (concave) function

    Concave down (concave) function
    The function is concave down (concave) in every interval of its domain where the sign of its second derivative is negative
  • Period: 1000 to 1099

    Classifying inflection points

    Determine the type of the points whhere the graph changes its concavity type
  • 1025

    Points of inflection

    Points of inflection
    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)
  • 1050

    Ascending point of inflection

    Ascending point of inflection
    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
  • 1075

    Descending point of inflection

    Descending point of inflection
    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
  • 1100

    Drawing the probable graph

    Drawing the probable graph
    All the information collected during the study of the function allows to sketch a probable graph representing geometrically the main properties of the function