Formulas

Calculus

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Functions and Their Graphs

Functions: \(f\), \(g\), \(y\), \(u\)
Argument (independent variable): \(x\)
Set of natural numbers: \(\mathbb{N}\)
Set of real numbers: \(\mathbb{R}\)
The base of natural logarithms: \(e\)
Natural numbers: \(n\)
Integers: \(k\)
Real numbers: \(a\), \(b\), \(c\), \(d\)
Angle: \(\alpha\)
Period of a function: \(T\)
  1. The concept of function is one of the most important in mathematics. It is defined as follows. Let two sets \(X\) and \(Y\) be given. If for every element \(x\) in the set \(X\) there is exactly one element (an image) \(y = f\left( x \right)\) in the set \(Y\), then it is said that the function \(f\) is defined on the set \(X\). The element \(x\) is called the independent variable, and respectively, the output \(y\) of the function is called the dependent variable. If we consider the number sets \(X \subset \mathbb{R}\), \(Y \subset \mathbb{R}\) (where \(\mathbb{R}\) is the set of real numbers), then the function \(y = f\left( x \right)\) can be represented as a graph in a Cartesian coordinate system \(Oxy\).
  2. Even function
    \(f\left( { – x} \right) = f\left( x \right)\)
  3. Odd function
    \(f\left( { – x} \right) = -f\left( x \right)\)
  4. Periodic function
    \(f\left( {x + kT} \right) = f\left( x \right),\)
    where \(k\) is an integer, \(T\) is the period of the function.
  5. Inverse function
    Given a function \(y = f\left( x \right)\). To find its inverse function of it, it is necessary solve the equation \(y = f\left( x \right)\) for \(x\) and then switch the variables \(x\) and \(y\). The inverse function is often denoted as \(y = {f^{ – 1}}\left( x \right)\). The graphs of the original and inverse functions are symmetric about the line \(y = x\).
  6. Inverse function
  7. Composite function
    Suppose that a function \(y = f\left( u \right)\) depends on an intermediate variable \(u\), which in turn is a function of the independent variable \(x\): \(u = g\left( x \right)\). In this case, the relationship between \(y\) and \(x\) represents a “function of a function” or a composite function, which can be written as \(y = f\left( {g\left( x \right)} \right)\). The two-layer composite functions can be easily generalized to an arbitrary number of “layers”.
  8. Linear function
    \(y = ax + b,\) \(x \in \mathbb{R}\).
    Here the number \(a\) is called the slope of the straight line. It is equal to the tangent of the angle between the straight line and the positive direction of the \(x\)-axis: \(a = \tan \alpha\). The number \(b\) is the \(y\)-intercept.
  9. Linear function
  10. Quadratic function
    The simplest quadratic function has the form
    \(y = x^2,\) \(x \in \mathbb{R}.\)
    In general, a quadratic function is described by the formula
    \(y = ax^2 + bx + c,\) \(x \in \mathbb{R},\)
    where \(a\), \(b\), \(c\) are real numbers (in this case \(a \ne 0.\)) The graph of a quadratic function is called a parabola. The direction of the branches of the parabola depends on the sign of the coefficient \(a\). If \(a \gt 0\), the parabola is concave upwards. If \(a \lt 0\), the parabola is concave downwards.
  11. Quadratic function
    Quadratic functions at different coefficients a
  12. Cubic function
    The simplest cubic function is given by
    \(y = x^3,\) \(x \in \mathbb{R}.\)
    In general, a cubic function is described by the formula
    \(y = ax^3 + bx^2 + cx + d,\) \(x \in \mathbb{R},\)
    where \(a\), \(b\), \(c\), \(d\) are real numbers \(\left({a \ne 0}\right).\) The graph of a cubic function is called a cubic parabola. When \(a \gt 0\), the cubic function is increasing, and when \(a \lt 0,\) the cubic function is, respectively, decreasing.
  13. Cubic function
  14. Power function
    \(y = x^n,\) \(x \in \mathbb{R},\) \(n \in \mathbb{N}\).
  15. Power functions for even powers
    Power functions for odd powers
  16. Square root function
    \(y = \sqrt x ,\) \(x \in \left[ {0,\infty } \right).\)
  17. Square root function
  18. Exponential functions
    \(y = a^x,\) \(x \in \mathbb{R},\) \(a \gt 0,\) \(a \ne 1,\)
    \(y = e^x\) when \(a = e \approx 2.71828182846\ldots\)
    An exponential function increases when \(a \gt 1\) and decreases when \(0 \lt a \lt 1\).
  19. Exponential functions
  20. Logarithmic functions
    \(y = {\log_a}x,\) \(x \in \left( {0,\infty } \right),\) \(a \gt 0,\) \(a \ne 1,\)
    \(y = \ln x\), when \(a = e,\;x \in \left( {0,\infty } \right).\)
    A logarithmic function increases if \(a \gt 1\) and decreases if \(0 \lt a \lt 1\).
  21. Logarithmic functions
  22. Hyperbolic sine function
    \(y = \sinh x =\) \({\large\frac{{{e^x} – {e^{ – x}}}}{2}\normalsize},\) \(x \in \mathbb{R}.\)
  23. Hyperbolic sine function
  24. Hyperbolic cosine function
    \(y = \cosh x =\) \({\large\frac{{{e^x} + {e^{ – x}}}}{2}\normalsize},\) \(x \in \mathbb{R}.\)
  25. Hyperbolic cosine function
  26. Hyperbolic tangent function
    \(y = \tanh x =\) \({\large\frac{{\sinh x}}{{\cosh x}}\normalsize} =\) \({\large\frac{{{e^x} – {e^{ – x}}}}{{{e^x} + {e^{ – x}}}}\normalsize},\) \(x \in \mathbb{R}.\)
  27. Hyperbolic tangent function
  28. Hyperbolic cotangent function
    \(y = \coth x =\) \({\large\frac{{\cosh x}}{{\sinh x}}\normalsize} =\) \({\large\frac{{{e^x} + {e^{ – x}}}}{{{e^x} – {e^{ – x}}}}\normalsize},\) \(x \in \mathbb{R},\) \(x \ne 0.\)
  29. Hyperbolic cotangent function
  30. Hyperbolic secant function
    \(y = {\mathop{\rm sech}\nolimits}\,x =\) \({\large\frac{1}{{\cosh x}}\normalsize} =\) \({\large\frac{2}{{{e^x} + {e^{ – x}}}}\normalsize},\) \(x \in \mathbb{R}.\)
  31. Hyperbolic secant function
  32. Hyperbolic cosecant function
    \(y = {\mathop{\rm csch}\nolimits}\,x =\) \({\large\frac{1}{{\sinh x}}\normalsize} =\) \({\large\frac{2}{{{e^x} – {e^{ – x}}}}\normalsize},\) \(x \in \mathbb{R},\) \(x \ne 0.\)
  33. Hyperbolic cosecant function
  34. Inverse hyperbolic sine function
    \(y = {\mathop{\rm arcsinh}\nolimits}\,x,\) \(x \in \mathbb{R}.\)
  35. Inverse hyperbolic sine function
  36. Inverse hyperbolic cosine function
    \(y = {\mathop{\rm arccosh}\nolimits}\,x,\) \(x \in \left[ {1,\infty } \right).\)
  37. Inverse hyperbolic cosine function
  38. Inverse hyperbolic tangent function
    \(y = {\mathop{\rm arctanh}\nolimits}\,x,\) \(x \in \left( {-1,1} \right).\)
  39. Inverse hyperbolic tangent function
  40. Inverse hyperbolic cotangent function
    \(y = {\mathop{\rm arccoth}\nolimits}\,x,\) \(x \in \left( { – \infty , – 1} \right) \cup \left( {1,\infty } \right).\)
  41. Inverse hyperbolic cotangent function
  42. Inverse hyperbolic secant function
    \(y = {\mathop{\rm arcsech}\nolimits}\,x,\) \(x \in \left( {0,1} \right].\)
  43. Inverse hyperbolic secant function
  44. Inverse hyperbolic cosecant function
    \(y = {\mathop{\rm arccsch}\nolimits}\,x,\) \(x \in \mathbb{R},\;x \ne 0.\)
  45. Inverse hyperbolic cosecant function