funkcje

Inputs, outputs, and values

A function assigns exactly one output $y$ to each permitted input $x$, and is written $y=f(x)$. The input is the argument; $f(x)$ is the function's value at that input.

Domain, range, and zeros

The Domain is the set of inputs for which a function is defined. The Range is the set of values the function can take. For example, for $f(x)=x^2$, the domain is $\mathbb{R}$ and the range is $[0,\infty)$.

To calculate a Function value, substitute the specified input for $x$. For example, if $f(x)=2x+3$, then

A Zero of a function is an input for which $f(x)=0$. For example, solving $2x-6=0$ gives the zero $x=3$.

Linear and quadratic functions

A Linear function has the form

Its graph crosses the $y$-axis at $(0,b)$. When $a\ne0$, its zero is $x_0=-\frac{b}{a}$. The value of $a$ determines its Monotonicity: if $a>0$, the function is increasing; if $a<0$, it is decreasing; and if $a=0$, it is constant.

A Quadratic function has the form

Its graph is a parabola. The discriminant and vertex are calculated as follows:

The discriminant determines the number of real zeros: $\Delta>0$ gives two, $\Delta=0$ gives one, and $\Delta<0$ gives none. If $a>0$, the parabola opens upward and has a minimum; if $a<0$, it opens downward and has a maximum.

Reading signs and behavior from graphs

The Sign of a function can be read from its graph: above the $x$-axis the function is positive, below it the function is negative, and on the axis its values are zero. Thus, the points where the graph meets the $x$-axis correspond to zeros. A graph can also be used to read zeros, sign, and monotonicity.

For monotonicity, an increasing function has larger values as $x$ increases; a decreasing function has smaller values as $x$ increases; and a constant function's value does not change.

Transforming graphs

The following Graph transformations change a graph's position or orientation:

• $f(x)+2$: shift upward by $2$; $f(x)-2$: shift downward by $2$.

• $f(x-2)$: shift right by $2$; $f(x+2)$: shift left by $2$.

• $-f(x)$: reflect across the $x$-axis; $f(-x)$: reflect across the $y$-axis.

Even and odd functions

An Even function satisfies $f(-x)=f(x)$, which means its graph is symmetric about the $y$-axis. An Odd function satisfies $f(-x)=-f(x)$, which means its graph is symmetric about the origin.

Other function families

Several other common function forms are:

• The Absolute value function: $f(x)=|x|$.

• The Square-root function: $f(x)=\sqrt{x}$.

• Inverse proportionality: $f(x)=\frac{a}{x}$, where $a\ne0$.

• The Exponential function: $f(x)=a^x$, where $a>0$ and $a\ne1$.

Checking allowed inputs

Use the function's form to determine which inputs are allowed. A polynomial has domain $\mathbb{R}$. For a fraction, the denominator cannot be zero. For an even-degree root, the expression under the root must be at least zero; if that root is in a denominator, the expression under it must be greater than zero. A logarithm's argument must be greater than zero.