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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