Functions

Definition

1.Suppose $A$ and $B$ are non-empty subsets of the real numbers. The elements of $A$ are the inputs, while the elements of $B$ are the possible outputs. A function $f$ gives each $x \in A$ one and only one value $f(x) \in B.$ We write:

$$ f \colon A \to B $$

3.The linear rule $2x - 1$ defines the function:

$$ f \colon \mathbb{R} \to \mathbb{R}, \quad f(x) = 2x - 1 $$

4.Writing $x \mapsto 2x - 1$ specifies the same assignment without naming the function. At the input $3,$ its value is:

$$ f(3) = 2 \cdot 3 - 1 = 5 $$

5.The symbol $f$ denotes the complete assignment, whereas $f(3)$ is the image of the particular input $3.$

6.

Uniqueness is required for each fixed input. Different inputs may have the same image.

7.A function $f \colon A \to B$ may have the following properties:

9.These properties are treated in detail in the corresponding entry on injective, surjective and bijective functions.

10.Every set $A$ has an identity function. Its value at any $x \in A$ is simply $x$ itself:

$$ \mathrm{id}_A \colon A \to A, \quad \mathrm{id}_A(x) = x $$

11.It is bijective, since distinct inputs give distinct outputs and every element of $A$ is attained. For a fixed value $c \in B,$ the constant function from $A$ to $B$ is defined by:

$$ f \colon A \to B, \quad f(x) = c $$

12.A constant function is not injective when $A$ has more than one element, and it is surjective exactly when $B = \{\ c \ \}.$

13.Given $E \subseteq A,$ the restriction of $f \colon A \to B$ to $E$ is defined by:

$$ f|_E \colon E \to B, \quad f|_E(x) = f(x) $$

14.Restricting the domain can make a non-injective function injective. If $f|_E$ is injective, the same assignment with codomain $f(E)$ defines a bijection from $E$ to $f(E)$ and hence has an inverse function.

15.The graph of $f$ is the set of ordered pairs $(x, f(x))$ obtained by pairing each input with its assigned output:

$$ G_f = \{\ (x, f(x)) \mid x \in A \ \} $$

16.Thus $G_f \subseteq A \times B,$ and each $x \in A$ occurs as the first coordinate of exactly one pair.

What is not a function

17.A relation $R \subseteq A \times B$ is a function from $A$ to $B$ only when every $x \in A$ is associated with exactly one value $y \in B.$

18.In the diagram, a single element $x_0$ corresponds to two distinct values in the codomain, so the relation is not a function. Formally, the required existence and uniqueness condition is:

$$ \forall \ x \in A,\ \exists! \ y \in B\ \vert \ (x, y) \in R $$

19.In the relation shown, both $(x_0, y_1)$ and $(x_0, y_2)$ belong to $R,$ with $y_1 \neq y_2,$ which violates uniqueness. The same failure occurs in the table:

20.The relation is not a function because $x = -3$ is associated with both $7$ and $10,$ rather than with exactly one value.

21.A criterion for deciding whether a curve in the plane is the graph of a function on its projection onto the $x$-axis is the vertical line test, which is passed exactly when every vertical line meets the curve in at most one point.

22.The curve on the left, a parabola, is the graph of a function, since each $x$ corresponds to exactly one $y.$ The curve on the right is not the graph of a function, since $x_2$ corresponds to more than one value of $y.$

Difference between codomain and range

23.For a function $f \colon A \to B,$ the codomain is the declared target set $B.$

24.The range of $f,$ also called the image of the function, is the set $f(A)$ of attained values. It is always a subset of $B.$

25.Consider the function $f \colon \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2.$ Its codomain is $\mathbb{R},$ while its range is $[0, +\infty),$ so it is not surjective. The same assignment with codomain $[0, +\infty)$ defines a surjective function, but it is not injective because $f(-1) = f(1) = 1.$ With both domain and codomain equal to $[0, +\infty),$ the same rule defines a bijection.

26.For a subset $E \subseteq A,$ its image is defined by:

$$ f(E) = \{\ f(x) \mid x \in E \ \} $$

27.Taking $E = A$ gives the range $f(A).$ For a subset $F \subseteq B,$ its preimage is defined by:

$$ f^{-1}(F) = \{\ x \in A \mid f(x) \in F \ \} $$

28.The notation $f^{-1}(F)$ denotes a set and does not require $f$ to be invertible. The function $f \colon A \to f(A)$ is surjective because every element of $f(A)$ is attained by construction.

Function equality and zeros

29.Under the convention used here, the domain and codomain are part of the data of a function. Two functions are equal when these sets agree and the functions have the same value at every point. For $f,g \colon D \to B,$ the pointwise condition is:

$$ f(x) = g(x) \quad \forall \ x \in D $$

30.A point $a \in D$ is a zero of $f$ if the function vanishes at that point:

$$ f(a) = 0 $$

31.Finding the zeros amounts to solving the equation $f(x)=0.$ Each solution $a$ identifies a point $(a,0)$ where the graph meets the $x$-axis. In sign analysis, every zero must be examined as a possible boundary between intervals on which $f$ is positive and those on which it is negative. The sign need not change at a zero, since the graph may touch the axis and remain on the same side.

Symmetric and bounded functions

32.Even and odd functions describe the behavior of a function under the change $x \mapsto -x.$ Let $A \subseteq \mathbb{R}$ be symmetric with respect to the origin, meaning $x \in A \Rightarrow -x \in A.$ For a function $f \colon A \to \mathbb{R},$ the two cases are:

34.A function $f \colon A \to \mathbb{R}$ with $A \subseteq \mathbb{R}$ is bounded if there exist $m, M \in \mathbb{R}$ such that:

$$ m \leq f(x) \leq M \quad \text{for every } x \in A $$

35.The function is bounded above if there exists $M \in \mathbb{R}$ such that $f(x) \leq M$ for every $x \in A,$ and bounded below if there exists $m \in \mathbb{R}$ such that $m \leq f(x)$ for every $x \in A.$

36.Boundedness means that the range lies in some bounded interval $[m, M].$ It does not imply the existence of a global maximum or minimum. A global maximum is an upper bound attained by the function, and a global minimum is a lower bound attained by the function (the arctangent is an example).

Monotone and periodic functions

37.Increasing, decreasing, and monotone functions compare function values at ordered points. For a function $f \colon A \to \mathbb{R},$ the inequality in each of the first four definitions below must hold for every pair $x_1, x_2 \in A$ with $x_1 < x_2:$

39.A function $f \colon X \to \mathbb{R}$ is periodic if there exists $T > 0$ for which the translation $x \mapsto x + T$ maps $X$ onto itself and the following identity holds for every $x \in X:$

$$ f(x + T) = f(x) $$

40.Every positive $T$ satisfying these conditions is a period of $f.$ If the set of positive periods has a least element, that element is the fundamental period. The sine and cosine functions both have fundamental period $2\pi.$

Algebraic and transcendental functions

41.One way to classify continuous real functions is through polynomial relations. Let $f \colon D \to \mathbb{R},$ where $D \subseteq \mathbb{R}$ contains a non-degenerate interval. The function is algebraic if there is a nonzero polynomial $P$ in two variables, with real coefficients and positive degree in $y,$ such that $P(x, f(x)) = 0$ for every $x \in D.$ For example, $f(x) = \sqrt{x}$ is algebraic on $[0, +\infty)$ because $f(x)^2 - x = 0.$

42.Every polynomial function is rational, since it can be written with denominator $1,$ and every rational function $f(x) = N(x)/Q(x)$ is algebraic because $Q(x)f(x) - N(x) = 0.$ Hence:

$$ \text{polynomial functions} \subseteq \text{rational functions} \subseteq \text{algebraic functions} $$

43.The irrational functions considered here form a subclass of the non-rational algebraic functions. Their simplified expressions contain the variable in a radical and use finitely many root extractions. They do not exhaust the algebraic functions, since some algebraic functions cannot be expressed by radicals.

44.A continuous real function that satisfies no such polynomial relation is transcendental. Standard elementary examples include exponential and logarithmic functions, trigonometric and inverse trigonometric functions, and hyperbolic functions. In this context, transcendental means non-algebraic, not non-elementary.

Domain of the main functions

45.The natural domain of a real expression is the largest subset of $\mathbb{R}$ on which every operation in the expression is defined and real-valued. When an expression imposes several restrictions, its natural domain is obtained by combining all domain conditions.

46.Polynomial functions have the form:

$$ y = \sum_{k=0}^{n} a_kx^k $$

47.The coefficients $a_0, a_1, \dots, a_n$ are real, and $n$ is a non-negative integer. When the polynomial has degree $n,$ its leading coefficient is $a_n,$ with $a_n \neq 0.$ The zero polynomial has all coefficients equal to zero. The monomial $x^0$ is the constant monomial $1.$ The constant term and the positive integer powers are defined for every real $x,$ including $x = 0.$ Hence every polynomial has natural domain $\mathbb{R}.$ Consider the cubic polynomial:

$$ y = 2x^3 - 5x^2 + 3x - 1 $$

48.Every term is defined for every real $x,$ so this polynomial also has domain $\mathbb{R}.$

49.In a rational function, the numerator $N(x)$ and denominator $D(x)$ are polynomials, and $D$ is not the zero polynomial. Rational functions have the form:

$$ y = \frac{N(x)}{D(x)} $$

50.The quotient is defined at exactly those real numbers $x$ for which $D(x) \neq 0.$ Consider the rational function:

$$ y = \frac{x^2 - 4}{x - 2} $$

51.The denominator vanishes at $x = 2,$ so the domain is $\mathbb{R} \setminus \{\ 2 \ \}.$ Although the expression simplifies to $x + 2$ when $x \neq 2,$ this cancellation does not add $2$ to the original domain.

52.Let $f \colon D \to \mathbb{R}$ and let $n \geq 2$ be an integer. A radical expression with index $n$ has the form:

$$ y = \sqrt[n]{f(x)} $$

53.The domain depends on the parity of $n.$ If $n$ is even, the radicand must be non-negative, so the domain is:

$$ \{\ x \in D \mid f(x) \geq 0 \ \} $$

54.For the even index $n = 2,$ consider:

$$ y = \sqrt{x - 2} $$

55.The radicand must be non-negative, so the domain is $[2, +\infty).$ If $n$ is odd, the radical imposes no further restriction, and the domain is $D.$ For the odd index $n = 3,$ consider:

$$ y = \sqrt[3]{x - 2} $$

56.The cube root is defined for negative and non-negative radicands, so its domain is $\mathbb{R}.$

57.For $f \colon D \to \mathbb{R},$ an expression involving a logarithm has the form:

$$ y = \log_a{f(x)} \quad \text{with} \quad a > 0, \quad a \neq 1 $$

58.The logarithmic argument must be strictly positive, so the domain is:

$$ \{\ x \in D \mid f(x) > 0 \ \} $$

59.For base $2,$ consider:

$$ y = \log_2(x - 1) $$

60.This function is defined only when $x - 1 > 0,$ so the domain is $(1, +\infty).$ For any $x \leq 1$ the expression is undefined, because the logarithm of a non-positive number does not exist in the reals. A second example uses the natural logarithm:

$$ y = \ln(3x + 6) $$

61.The inequality $3x + 6 > 0$ gives $x > -2,$ so the domain is $(-2, +\infty).$

62.For $f \colon D \to \mathbb{R},$ an exponential expression with constant base has the form:

$$ y = a^{f(x)} \quad \text{with} \quad a > 0, \quad a \neq 1 $$

63.Its domain is $D.$ Taking $a = 2$ and $f(x) = x$ gives:

$$ y = 2^x $$

64.Since $2 > 0,$ this function has domain $\mathbb{R}$ and range $(0, +\infty).$

65.Expressions with a variable base and exponent have the form:

$$ y = [f(x)]^{g(x)} $$

66.If real powers are defined by $[f(x)]^{g(x)} = \exp(g(x)\ln f(x)),$ then $f(x) > 0$ is required. Under this definition, if $f$ and $g$ have domains $D_f$ and $D_g,$ respectively, the domain is:

$$ \{\ x \in D_f \cap D_g \mid f(x) > 0 \ \} $$

67.For non-positive bases, the real domain depends on the values of the exponent and must be determined case by case.

68.For $f \colon D \to \mathbb{R},$ a power with an irrational exponent $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ has the form:

$$ f(x)^{\alpha} $$

69.Its domain is given by the following two cases:

70.$$ \{\ x \in D \mid f(x) \geq 0 \ \}, \quad \text{if } \alpha > 0 $$ $$ \{\ x \in D \mid f(x) > 0 \ \}, \quad \text{if } \alpha < 0 $$

Operations between functions

72.Let $f \colon X_1 \to \mathbb{R}$ and $g \colon X_2 \to \mathbb{R},$ where $X_1 \subseteq \mathbb{R}$ and $X_2 \subseteq \mathbb{R},$ and set $X=X_1 \cap X_2.$

73.The sum, difference, and product are defined pointwise on $X:$

$$ \begin{align} (f + g)(x) &= f(x) + g(x) \\[6pt] (f - g)(x) &= f(x) - g(x) \\[6pt] (f \cdot g)(x) &= f(x)g(x) \end{align} $$

74.The quotient of two functions $f(x)$ and $g(x)$ has domain $\{\ x \in X \mid g(x) \neq 0 \ \}$ and is defined by:

$$ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} $$

75.Another operation between functions is composition. If $f \colon A \to B$ and $g \colon B \to C,$ then the composite $g \circ f$ applies $f$ first and then applies $g$ to the result:

$$ (g \circ f)(x)=g(f(x)) $$