Function features

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A function f of a real variable x is a rule that assigns to each x ∈ R at most one y ∈ R such that
y = f(x)

The domain of the function D is the set of all x ∈ R for which there exists exactly one y ∈ R such that
y = f(x).

The range of the function H is the set of all y ∈ R for which there exists at least one x ∈ R such that
y = f(x).

x2​>x1​,f(x2​)>f(x1​), the function is increasing
$x_2 > x_1, f(x_2) < f(x_1)$, the function is decreasing
$x_2 > x_1, f(x_2) \leq f(x_1)$, the function is non-increasing
$x_2 > x_1, f(x_2) \geq f(x_1)$, the function is non-decreasing
$x_2 \neq x_1, f(x_2) \neq f(x_1)$, the function is injective

If there exists $h\in$, such that $f(x)\le h$, the function is bounded from above
If there exists $d\in$R, such that $f(x) \g$, the function is bounded from below
If the function is bounded from above and from below, it is bounded

If $f(x + kp) = f(x)$, the function is periodic ($p =$ period)
If $f(-x)=f$, the function is even
If $f(-x) = -f(x)$, the function is odd

A function $f(x)$ has an inverse function $f^{-1}(x)$. It holds:
$D^{-}=H,H^{-}=D$

Two functions are equal: f(x) = g(x) if: D(f) = D(g)

                                                       f(x) = g(x)

Which of the following graphs represent functions?

Function                                                  Function                                         Not a function 

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For the given functions apply: f₃(x) = F₄(X)

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The solution to the system is a=2, b=5, c=-3

f: y=2x² + 5x -3 

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The function is decreasing and therefore holds

The set consists of all functions f(x): y= 2/(x-a)

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The equations of the functions are: f₁: y = – x + 2 and f₂: y = √3.

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x = speed,  y = consumption

Consumption at a speed of 90 km/h.

The fuel consumption of the car at a speed of 90 km/h is 6.7 liters per 100 kilometers.

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The quadratic function has the equation: f : y = x² - 3x +1

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With the x-axis: y = 0                             With the y-axis : x = 0

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The equation of the logarithmic function is: y = log x + 2.

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The function has the form f: y = 2^x − 1.

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