A function is one of the most important concepts in mathematics. Functions describe dependencies between quantities and form the basis for studying algebra, analysis, and many other branches. On the YÖS exam, problems about functions make up a significant portion, testing understanding of the concept of a function, its properties, and ways of defining it.
A function is a rule that assigns to each element from one set (the domain) exactly one element from another set (the range).
Analogy: If you think of a function as a machine:- A number from the domain is supplied as input- The machine processes this number according to a certain rule- The output is exactly one number from the range
Definition: A function $f: A \to B$ is a relation between sets $A$ and $B$ such that each element $x \in A$ is associated with exactly one element $y \in B$.
Formally: $f$ is a function $\Leftrightarrow$1. $\forall x \in A: \exists y \in B$ such that $(x, y) \in f$2. $\forall x \in A: \forall y_1, y_2 \in B$ if $(x, y_1) \in f$ and $(x, y_2) \in f$, then $y_1 = y_2$
Notation:- $f: A \to B$ (function from $A$ to $B$)- $f(x) = y$ (the function $f$ maps $x$ to $y$)- $x \mapsto f(x)$ (mapping rule)
Domain: The set $A$ on which the function $f$ is defined.
Notation: $D(f)$, $Dom(f)$, or simply $A$
Codomain: The set $B$ into which the function $f$ maps.
Notation: $B$ or $Cod(f)$
Image (Range/Image): The set of all values that the function actually takes.
Notation: $E(f)$, $Im(f)$, $f(A)$ or $\{f(x) \mid x \in A\}$
Important remark: The image of a function is often smaller than the codomain. The codomain is the "target set", while the image is the "set into which the function actually maps elements".
Example 1: Consider the function $f(x) = x^2$ on the set of real numbers.
A function is specified by means of a mathematical formula.
Examples:- $f(x) = 3x + 2$- $g(x) = x^2 - 4x + 1$- $h(x) = \sqrt{x - 1}$
Advantages: Compact, easy to compute values, properties can be analyzed.
Disadvantages: It may be difficult to visualize the behavior of the function.
A function is represented on the coordinate plane, where the values of $x$ are plotted on the horizontal axis and the values of $f(x)$ on the vertical axis.
Examples:- A straight line for a linear function- A parabola for a quadratic function- A sinusoid for the sine function
Advantages: Clearly shows the behavior of the function.
Disadvantages: Hard to compute exact values, requires drawing.
A function is given by a table that lists some values of $x$ and the corresponding values of $f(x)$.
Example:
| $x$ | 0 | 1 | 2 | 3 | 4 ||-----|---|---|---|---|---|| $f(x)$ | 5 | 7 | 9 | 11 | 13 |
Advantages: Convenient for a finite set of points, used in statistics and practical applications.
Disadvantages: Fully determines the function only on a finite set, requires a lot of space for a large number of points.
A function is described in words in natural language.
Example: "The function $f$ assigns to each person their date of birth."
Advantages: Can be used for functions that are hard to express by a formula.
Disadvantages: Less precise and formal.
Definition: A linear function is a function of the form:
$$f(x) = ax + b$$
where $a$ and $b$ are constants, $a \neq 0$.
Properties:- The graph is a straight line- Slope: $a$ (determines the inclination of the line)- Intercept: $b$ (point of intersection with the $y$-axis)- Monotonically increasing if $a > 0$- Monotonically decreasing if $a < 0$
Examples:- $f(x) = 2x + 3$- $g(x) = -x + 5$- $h(x) = \frac{1}{2}x - 1$
Definition: A quadratic function is a function of the form:
$$f(x) = ax^2 + bx + c$$
where $a$, $b$, $c$ are constants, $a \neq 0$.
Properties:- The graph is a parabola- Axis of symmetry: $x = -\frac{b}{2a}$- Vertex of the parabola: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$- If $a > 0$, the parabola opens upward (minimum at the vertex)- If $a < 0$, the parabola opens downward (maximum at the vertex)
Vertex form:$$f(x) = a(x - h)^2 + k$$where $(h, k)$ are the coordinates of the vertex
Examples:- $f(x) = x^2 - 4x + 3$- $g(x) = -2x^2 + 8x - 1$- $h(x) = (x - 1)^2 + 2$
Definition: A polynomial function of degree $n$ is a function of the form:
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$
where $a_n \neq 0$ and $n$ is a natural number.
Examples:- Degree 1: linear function- Degree 2: quadratic function- Degree 3: cubic function ($f(x) = x^3$)
Definition: A rational function is a function that is a ratio of two polynomials:
$$f(x) = \frac{P(x)}{Q(x)}$$
where $P(x)$ and $Q(x)$ are polynomials, $Q(x) \neq 0$.
Example:$$f(x) = \frac{x + 2}{x - 1}, \quad x \neq 1$$
Definition: A piecewise function is a function that is defined by different formulas on different parts of its domain.
Example:$$f(x) = \begin{cases}x + 1, & \text{if } x < 0 \\x^2, & \text{if } 0 \leq x \leq 2 \\4, & \text{if } x > 2\end{cases}$$
Definition of an even function: A function $f$ is called even if:1. Its domain is symmetric about zero2. $f(-x) = f(x)$ for all $x$ in the domain
Geometric meaning: The graph of an even function is symmetric about the $y$-axis.
Examples of even functions:- $f(x) = x^2$- $g(x) = |x|$- $h(x) = \cos(x)$
Definition of an odd function: A function $f$ is called odd if:1. Its domain is symmetric about zero2. $f(-x) = -f(x)$ for all $x$ in the domain
Geometric meaning: The graph of an odd function is symmetric about the origin.
Examples of odd functions:- $f(x) = x$- $g(x) = x^3$- $h(x) = \sin(x)$
Remark: A function may be neither even nor odd.
Definition of an increasing function: A function $f$ is called increasing on an interval $(a, b)$ if for any $x_1, x_2 \in (a, b)$ with the property $x_1 < x_2$ it holds that $f(x_1) < f(x_2)$.
Definition of a decreasing function: A function $f$ is called decreasing on an interval $(a, b)$ if for any $x_1, x_2 \in (a, b)$ with the property $x_1 < x_2$ it holds that $f(x_1) > f(x_2)$.
Examples:- $f(x) = 2x$ is increasing on the entire real line- $g(x) = -x + 3$ is decreasing on the entire real line- $h(x) = x^2$ is decreasing on $(-\infty, 0)$ and increasing on $(0, +\infty)$
Definition of a function bounded above: A function $f$ is bounded above on a set $D$ if there exists a number $M$ such that $f(x) \leq M$ for all $x \in D$.
Definition of a function bounded below: A function $f$ is bounded below on a set $D$ if there exists a number $m$ such that $f(x) \geq m$ for all $x \in D$.
Definition of a bounded function: A function $f$ is called bounded on a set $D$ if it is bounded both above and below.
Example:$$f(x) = \sin(x)$$Bounded: $-1 \leq \sin(x) \leq 1$ for all $x$
Definition: A function $f: A \to B$ is called an injection (or one-to-one), if different elements from $A$ are mapped to different elements in $B$.
Formally: $f$ is an injection $\Leftrightarrow \forall x_1, x_2 \in A: (x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2))$
Equivalently: $\forall x_1, x_2 \in A: (f(x_1) = f(x_2) \Rightarrow x_1 = x_2)$
Example: $f(x) = 2x + 1$ is an injection from $\mathbb{R}$ to $\mathbb{R}$
Counterexample: $f(x) = x^2$ is not an injection from $\mathbb{R}$ to $\mathbb{R}$, since $f(2) = f(-2) = 4$
Definition: A function $f: A \to B$ is called a surjection if every element of $B$ is the image of at least one element of $A$.
Formally: $f$ is a surjection $\Leftrightarrow \forall y \in B: \exists x \in A \text{ such that } f(x) = y$
Equivalently: $\text{Image}(f) = B$
Example: $f: \mathbb{R} \to [0, +\infty)$, $f(x) = x^2$ is a surjection
Definition: A function $f: A \to B$ is called a bijection if it is both an injection and a surjection.
Formally: $f$ is a bijection $\Leftrightarrow$ $f$ is injective AND $f$ is surjective
Example: $f: \mathbb{R} \to \mathbb{R}$, $f(x) = 2x + 1$ is a bijection
Importance: Only bijections have inverse functions.
Definition: The composition of functions $f: A \to B$ and $g: B \to C$ is the function $g \circ f: A \to C$ defined by:
$$(g \circ f)(x) = g(f(x))$$
Read as: "$g$ after $f$" or "composition of $g$ and $f$"
Order: First the function $f$ is applied, then $g$ is applied to the result.
Example 2: Let $f(x) = x + 1$ and $g(x) = 2x$. Find $(g \circ f)(x)$ and $(f \circ g)(x)$.
Solution:$$(g \circ f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2$$
$$(f \circ g)(x) = f(g(x)) = f(2x) = 2x + 1$$
Conclusion: $g \circ f \neq f \circ g$ — composition is not commutative!
Example 3: Find $(g \circ f)(3)$ if $f(x) = x^2$ and $g(x) = \sqrt{x}$, where $x \geq 0$.
Solution:$$(g \circ f)(3) = g(f(3)) = g(3^2) = g(9) = \sqrt{9} = 3$$
Definition: A function $f^{-1}: B \to A$ is called the inverse of the function $f: A \to B$ if:
$$(f^{-1} \circ f)(x) = x \text{ for all } x \in A$$$$(f \circ f^{-1})(y) = y \text{ for all } y \in B$$
Formally: $f^{-1}(y) = x \Leftrightarrow f(x) = y$
The inverse function exists if and only if $f$ is a bijection (injection and surjection at the same time).
Algorithm:1. Verify that the function is a bijection2. Denote $y = f(x)$3. Express $x$ in terms of $y$: $x = f^{-1}(y)$4. Swap the variable names: $y = f^{-1}(x)$
Example 4: Find the inverse function of $f(x) = 3x - 2$.
Solution:
Step 1: Check. $f$ is a linear function with $a = 3 \neq 0$, so $f$ is a bijection.
Step 2: $y = 3x - 2$
Step 3: Solve for $x$:$$y = 3x - 2$$$$y + 2 = 3x$$$$x = \frac{y + 2}{3}$$
Step 4: Swap notation:$$f^{-1}(x) = \frac{x + 2}{3}$$
Verification:$$(f \circ f^{-1})(x) = f(f^{-1}(x)) = f\left(\frac{x + 2}{3}\right) = 3 \cdot \frac{x + 2}{3} - 2 = x + 2 - 2 = x$$ ✓
Example 5: Find the inverse function of $f(x) = x^2$ on $[0, +\infty)$.
On the set $[0, +\infty)$, the function $f(x) = x^2$ is a bijection (it is increasing and covers $[0, +\infty)$).
Step 2: $y = x^2$ (where $x \geq 0$)
Step 3: $x = \sqrt{y}$ (we take the positive root, since $x \geq 0$)
Step 4: $f^{-1}(x) = \sqrt{x}$ on $[0, +\infty)$
Example 6: Find $f(2)$ and $f(-1)$ if $f(x) = x^2 - 3x + 2$.
Solution:$$f(2) = 2^2 - 3(2) + 2 = 4 - 6 + 2 = 0$$$$f(-1) = (-1)^2 - 3(-1) + 2 = 1 + 3 + 2 = 6$$
Example 7: Find the function $f(x)$ given that $f(x+1) = 2f(x) + 3$ and $f(0) = 1$.
We use the recurrence relation:- $f(0) = 1$- $f(1) = 2f(0) + 3 = 2(1) + 3 = 5$- $f(2) = 2f(1) + 3 = 2(5) + 3 = 13$- $f(3) = 2f(2) + 3 = 2(13) + 3 = 29$
Let us look for a pattern. This is a linear recurrence relation.
General solution: $f(x) = A \cdot 2^x + B$
From $f(0) = 1$: $A + B = 1$From $f(1) = 5$: $2A + B = 5$
Subtracting: $A = 4$, hence $B = -3$
Answer: $f(x) = 4 \cdot 2^x - 3 = 2^{x+2} - 3$
Determine whether the function $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^3 + 1$ is an injection, a surjection, or a bijection.
Injectivity: Assume $f(x_1) = f(x_2)$. Then:$$x_1^3 + 1 = x_2^3 + 1$$$$x_1^3 = x_2^3$$$$x_1 = x_2$$
The function is injective. ✓
Surjectivity: For any $y \in \mathbb{R}$ we need to find $x \in \mathbb{R}$ such that $f(x) = y$.$$x^3 + 1 = y$$$$x^3 = y - 1$$$$x = \sqrt[3]{y - 1}$$
Such an $x$ exists for any $y$. The function is surjective. ✓
Conclusion: The function $f$ is a bijection.
Let $f(x) = 2x + 1$ and $g(x) = x^2 - 3$. Find $(f \circ g)(2)$ and $(g \circ f)(2)$.
$$(f \circ g)(2) = f(g(2)) = f(2^2 - 3) = f(1) = 2(1) + 1 = 3$$
$$(g \circ f)(2) = g(f(2)) = g(2(2) + 1) = g(5) = 5^2 - 3 = 22$$
Answer: $(f \circ g)(2) = 3$, $(g \circ f)(2) = 22$
Find the inverse of $f(x) = \frac{2x - 1}{x + 3}$ on the domain where it is defined.
Step 1: Domain of $f$: $x \neq -3$
Step 2: $y = \frac{2x - 1}{x + 3}$
Step 3: Solve for $x$:$$y(x + 3) = 2x - 1$$$$xy + 3y = 2x - 1$$$$xy - 2x = -1 - 3y$$$$x(y - 2) = -1 - 3y$$$$x = \frac{-1 - 3y}{y - 2} = \frac{-(1 + 3y)}{y - 2}$$
Step 4: $f^{-1}(x) = \frac{-(1 + 3x)}{x - 2} = \frac{-1 - 3x}{x - 2}$, $x \neq 2$
Verification:$$(f \circ f^{-1})(x) = f\left(\frac{-1 - 3x}{x - 2}\right) = \frac{2 \cdot \frac{-1 - 3x}{x - 2} - 1}{\frac{-1 - 3x}{x - 2} + 3}$$
(After simplification, the result is $x$.)
Determine whether the function $f(x) = x^3 + x$ is even, odd, or neither.
Compute $f(-x)$:$$f(-x) = (-x)^3 + (-x) = -x^3 - x = -(x^3 + x) = -f(x)$$
Since $f(-x) = -f(x)$, the function $f$ is odd.
Given the function:$$f(x) = \begin{cases}2x, & \text{if } x < 1 \\3, & \text{if } 1 \leq x \leq 2 \\x + 1, & \text{if } x > 2\end{cases}$$
Find $f(0)$, $f(1)$, $f(1.5)$, $f(2)$, $f(3)$.
Answer: $f(0) = 0$, $f(1) = 3$, $f(1.5) = 3$, $f(2) = 3$, $f(3) = 4$
### Mistake 1: Confusing the domain and the codomainWrong: The range of the function $f(x) = x^2$ is $\mathbb{R}$.Correct: The range of the function $f(x) = x^2$ is $[0, +\infty)$. The domain can be $\mathbb{R}$.
### Mistake 2: Ignoring the conditions for the existence of the functionWrong: The function $f(x) = \frac{1}{x}$ is defined for all real numbers.Correct: The function $f(x) = \frac{1}{x}$ is defined for all $x \neq 0$.
### Mistake 3: Wrong order in the compositionWrong: $(g \circ f)(x) = f(g(x))$Correct: $(g \circ f)(x) = g(f(x))$
### Mistake 4: Confusing a function with a relationWrong: The relation $x = y^2$ defines a function from $\mathbb{R}$ to $\mathbb{R}$.Correct: This is a relation, not a function (one value of $x$ corresponds to two values of $y$).
### Mistake 5: Misapplication of the definition of injectionWrong: A function is injective if $f(x_1) \neq f(x_2)$ for all $x_1 \neq x_2$.Correct: A function is injective if $f(x_1) = f(x_2)$ implies $x_1 = x_2$.
### Mistake 6: Forgetting the property $(f^{-1})' = 1/f'$ when finding the derivative of the inverse function (for advanced level)Wrong: The derivative of the inverse function equals the derivative of the original function.Correct: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$
### Mistake 7: Incorrect definition of evenWrong: A function is even if it has an even degree (for example, $x^2$).Correct: A function is even if $f(-x) = f(x)$ for all $x$ in the domain.
### Tip 1: Checking injectivity and surjectivity graphically- A function is injective if every horizontal line intersects the graph at most once (horizontal line test)- A function is surjective onto a given set if its graph covers every value of that set
### Tip 2: Visualizing compositionThink of composition as a "chain" of transformations:$$x \xrightarrow{f} f(x) \xrightarrow{g} g(f(x))$$
### Tip 3: Preserving the order when finding an inverse functionAlways check that the function is a bijection before finding its inverse.
### Tip 4: Taking the domain into account when working with functionsAlways keep in mind the restrictions on the domain:- You cannot divide by zero- You cannot take the square root of a negative number (within the real numbers)- The logarithm is defined only for positive numbers
### Tip 5: Checking even and oddCompute $f(-x)$ and compare it with $f(x)$ and $-f(x)$:- If $f(-x) = f(x)$ → even- If $f(-x) = -f(x)$ → odd- Otherwise → neither even nor odd
### Tip 6: Working with piecewise functionsSketch the graph of a piecewise function to understand its behavior. Pay attention to the "gluing" points of the different pieces.
### Tip 7: Applying composition in problemsWhen you see "find the value of $f(g(x))$", compute step by step:1. First find $g(x)$2. Then substitute this result into $f$
### Tip 8: Verifying the result of an inverse functionAlways verify the inverse function you have found by applying the composition $(f \circ f^{-1})(x) = x$.
### Tip 9: Practice with concrete numbersIf you are not sure about the result when working with abstract functions, check it on concrete numerical examples.
### Tip 10: Learn the standard functionsMemorize the properties of frequently occurring functions:- Linear functions (always bijections)- Quadratic functions (neither injections nor surjections on $\mathbb{R}$)- Cubic functions (always bijections)- Exponential functions (injections, but not surjections onto $\mathbb{R}$)- Logarithmic functions (injections, surjections onto their range)
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Conclusion
Functions are a central concept of mathematics, and a deep understanding of them is critically important for success on the YÖS exam. Practice various types of problems until working with functions becomes second nature.
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