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Calculus

# 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.
1. 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 $$y = x$$.
1. 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”.
1. 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.
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.
1. 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.
1. Power function
$$y = x^n,$$ $$x \in \mathbb{R},$$ $$n \in \mathbb{N}$$.
1. Square root function
$$y = \sqrt x ,$$ $$x \in \left[ {0,\infty } \right).$$
1. 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 $$0 \lt a \lt 1$$.
1. 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 $$0 \lt a \lt 1$$.
1. Hyperbolic sine function
$$y = \sinh x =$$ $${\large\frac{{{e^x} – {e^{ – x}}}}{2}\normalsize},$$ $$x \in \mathbb{R}.$$
1. Hyperbolic cosine function
$$y = \cosh x =$$ $${\large\frac{{{e^x} + {e^{ – x}}}}{2}\normalsize},$$ $$x \in \mathbb{R}.$$
1. Hyperbolic tangent function
$$y = \tanh x =$$ $${\large\frac{{\sinh x}}{{\cosh x}}\normalsize} =$$ $${\large\frac{{{e^x} – {e^{ – x}}}}{{{e^x} + {e^{ – x}}}}\normalsize},$$ $$x \in \mathbb{R}.$$
1. Hyperbolic cotangent function
$$y = \coth x =$$ $${\large\frac{{\cosh x}}{{\sinh x}}\normalsize} =$$ $${\large\frac{{{e^x} + {e^{ – x}}}}{{{e^x} – {e^{ – x}}}}\normalsize},$$ $$x \in \mathbb{R},$$ $$x \ne 0.$$
1. Hyperbolic secant function
$$y = {\mathop{\rm sech}\nolimits}\,x =$$ $${\large\frac{1}{{\cosh x}}\normalsize} =$$ $${\large\frac{2}{{{e^x} + {e^{ – x}}}}\normalsize},$$ $$x \in \mathbb{R}.$$
1. Hyperbolic cosecant function
$$y = {\mathop{\rm csch}\nolimits}\,x =$$ $${\large\frac{1}{{\sinh x}}\normalsize} =$$ $${\large\frac{2}{{{e^x} – {e^{ – x}}}}\normalsize},$$ $$x \in \mathbb{R},$$ $$x \ne 0.$$
1. Inverse hyperbolic sine function
$$y = {\mathop{\rm arcsinh}\nolimits}\,x,$$ $$x \in \mathbb{R}.$$
1. Inverse hyperbolic cosine function
$$y = {\mathop{\rm arccosh}\nolimits}\,x,$$ $$x \in \left[ {1,\infty } \right).$$
1. Inverse hyperbolic tangent function
$$y = {\mathop{\rm arctanh}\nolimits}\,x,$$ $$x \in \left( {-1,1} \right).$$
1. Inverse hyperbolic cotangent function
$$y = {\mathop{\rm arccoth}\nolimits}\,x,$$ $$x \in \left( { – \infty , – 1} \right) \cup \left( {1,\infty } \right).$$
1. Inverse hyperbolic secant function
$$y = {\mathop{\rm arcsech}\nolimits}\,x,$$ $$x \in \left( {0,1} \right].$$
1. Inverse hyperbolic cosecant function
$$y = {\mathop{\rm arccsch}\nolimits}\,x,$$ $$x \in \mathbb{R},\;x \ne 0.$$