Study of a function

Determining the domain

Period: 100 to 199

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

Solve inequalities

150

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, ...)

Determining symmetries

Period: 200 to 299

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

Parity (axial / central symmetry)

233

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

Periodicity (translation)

267

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

Determining sign and intersections of the graph with coordinate axes

Period: 300 to 399

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

Intersection with the axis of ordinates

325

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

Intersections with the axis of abscissas

350

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

Sign of the function

375

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

Classifying discontinuities/singularities

Period: 400 to 499

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

Jump discontinuty / singularity (of the first kind)

425

The left and right limits are different real numbers

Essential discontiunity / singularity (of the second kind)

450

At least one among the left and right limits is infinite or does not exist

Removable discontinuity / singularity (of the third kind)

475

The left and the right limit are the same real number

Determining asymptotes

Period: 500 to 599

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

Vertical asymptote

525

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

Classifying inflection pointsHorizontal asymptote

550

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

Slant asymptote

575

For a slant asymptote to the graph of a function to exists three conditions need to be met

Determining growth (monotonicity)

Period: 600 to 699

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

Calculate the (first) derivative

625

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

Increasing function

650

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

Decreasing function

675

The function is decreasing in every interval of its domain where the sign of its (first) derivative is negative

Classifying stationary (critical) points

Period: 700 to 799

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

Critical points

720

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)

Local maximum

740

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

Local minimum

760

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

Horizontal point of inflection

780

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

Classifying points of non-derivability

Period: 800 to 899

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

Corner point

825

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

Cusp

850

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

Vertical point of inflection

875

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

Determining the concavity

Period: 900 to 999

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

Calculate the second derivative

925

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

Concave up (convex) function

950

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

Concave down (concave) function

975

The function is concave down (concave) in every interval of its domain where the sign of its second derivative is negative

Classifying inflection points

Period: 1000 to 1099

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

Points of inflection

1025

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)

Ascending point of inflection

1050

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

Descending point of inflection

1075

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

Drawing the probable graph

1100

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