Increasing and Decreasing Functions

As long as algebra is taught in school, there will be prayer in school, Cokie Roberts

Recall

Definition. A function f is a rule, relationship, or correspondence that assigns to each element x in a set D, x ∈ D (called the domain) exactly one element y in a set E, y ∈ E (called the codomain or range).

• The pair (x, y) is denoted as y = f(x) where: x is the independent variable (input) and y is the dependent variable (output). Often, both the domain D and codomain E are the set of real numbers ℝ or subsets of ℝ.

• D is the domain, the set of all possible inputs. E is the codomain or range, the set of all possible outputs.

• Key property💡: Each input has exactly one output. (No input is assigned two different outputs — this is the vertical line test!)

• Examples: constant, f(x) = c, horizontal line, slope = 0; linear, f(x) = mx + b, straight line, constant slope m and y-intercept b; quadratic $f(x) = ax^2 + bx + c$, u-shaped or inverted U, opens up (a > 0) or down (a < 0), vertex at x = $\frac{-b}{2a}$, symmetry about vertical axis through vertex; polynomial, $f(x) = a_n x^n + \dots + a_0$, a smooth and continuous curve, n roots (counting multiplicity), end behaviour determined by its leading term $a_n x^n$; exponential function, $f(x) = a \cdot b^x, a \ne 0, b \gt 0$, rapid growth (b > 1) or decay (0 < b < 1); trigonometric functions, $\sin(x), \cos(x), \tan (z)$ oscillatory, periodic behavior (period 2π for sin/cos, π for tan), sin and cos are bounded between -1 and 1, but tan is unbounded; step function $f(x) = \lfloor x \rfloor$, greatest integer ≤ x, constant on intervals [n, n+1), jumps at integers, its graph is a staircase shape; absolute value f(x) = |x|, V-shaped graph, slope changes at 0.

• Functions can be expressed in multiple forms, each useful in different contexts: verbal description, table of values (list of pairs), algebraic formula, graph, piecewise definition, recursive definition, parametric or integral form, and series representation.

• Evaluating a function means finding or computing the output value f(x) for a given input value x. f(x) = $x^2-2x +4, f(2) = 2^2 -2\cdot 2 + 4 = 4 - 4 + 4 = 4, f(0) = 0^2 -2\cdot 0 + 4 = 4, f(1) = 1^2 -2\cdot 1 + 4 = 1 -2 +4 = 3$

• The x-intercept is any point on the graph that intersects or crosses the x-axis. In other words, it is the value of x when the function (y-coordinate or y-value) is zero. The y-intercept is the point where the graph intersects or crosses the y-axis. y-coordinate of the point whose x-coordinate is 0, e.g., 2x - 3y = 6. x-intercept: set y = 0 → 2x = 6 ⇒ x=3, so (3, 0). y-intercept: set x = 0 → −3y = 6 ⇒ y = −2, so (0, −2).

• f is said to have a local or relative maximum at c if there exists an interval (a, b) containing c ($c \in (a, b)$) such that f(c) ≥ f(x) $∀ x \in (a, b) ∩ D$. f is said to have a local or relative minimum at c if there exists an interval (a, b) containing c ($c \in (a, b)$) such that f(c) ≤ f(x) $∀ x \in (a, b) ∩ D$.💡Local extrema can only occur where the function stops rising or falling —either because the derivative is zero, the derivative doesn't exist, or you're at the edge of the domain.

• First Derivative Test. Let f be differentiable on an open interval containing c, except possibly at c itself, and let c be a critical point (so f′(c) = 0 or f′ is undefined). Then:

1. If f′(x) > 0 for x just to the left of c and f′(x) < 0 for x just to the right of c, then f(c) is a local maximum.
2. If f′(x) < 0 for x just to the left of c and f′(x) > 0 for x just to the right of c, then f(c) is a local minimum.
3. If f′(x) does not change sign at c (stays positive or stays negative), then f(c) is not a local extremum.

• Second Derivative Test. If the first derivate of a function at a point c is zero (f'(c) = 0) and the second derivate at this point exists, then: (i) If f''(c) > 0, then f(c) is a local minimum. (ii) If f''(c) < 0, then f(c) is a local maximum. (iii) If f''(c) = 0, the test is inconclusive.
• Vertical asymptotes are vertical lines (perpendicular to the x-axis) of the form x = a (where a is a constant) near which the function grows without bound when x approaches a from at least one side. The line x = a is a vertical asymptote of f if at least one of the one-sided limits is infinite: $\lim_{x \to a^{-}}f(x)=\pm\infty$ or $\lim_{x \to a^{+}}f(x)=\pm\infty$.
• Horizontal asymptotes are horizontal lines (y = c, parallel to the x-axis) that the graph of the function approaches as x grows very large (positively or negatively). In other words, the function values get arbitrarily close to c as long as x is sufficiently large or more formally: $\lim_{x \to \infty}f(x)=c$ and/or $\lim_{x \to -\infty}f(x)=c$.
• A line y = m+b is an oblique (slant) asymptote as $x \to \infty$ if $\lim_{x\to\infty} [f(x) - (mx+b)] = 0.$ This means that the curve of f(x) gets more and more closer to the line y = mx + b as x grows large. For an asymptote y = mx + b as $x \to \infty, m = \lim_{x\to\infty} \frac{f(x)}{x}, b = \lim_{x\to\infty} \bigl(f(x) - m x\bigr)$, provided both limits exist.

Monotonic functions

A monotonic function is a mathematical function that maintains a consistent direction of change on its domain, either entirely increasing (monotonic increasing) or entirely decreasing (monotonic decreasing).

Let f be a function defined on some interval I. f increases on an interval I if f(b) >= f(a) ∀b > a, a,b ∈ I. If f(b) > f(a) ∀b > a, a, b ∈ I, the function is said to be strictly increasing on I.

Examples: f(x) = 2x, the exponential function f(x) = e^x, the natural logarithm function f(x) = ln(x), and the square root function f(x) = $\sqrt{x}$ (for x ≥ 0) are strictly increasing functions.

Conversely, f decreases on an interval I if f(b) <= f(a) ∀b > a, a, b ∈ I. If f(b) < f(a) ∀b > a, a, b ∈ I, the function is said to be strictly decreasing. Finally, f is said to be constant on an interval I if f(a) = f(b) ∀a, b ∈ I.

Examples. f(x) = -2x, f(x) = $-\sqrt{x}$, f(x) = 0.5^x (this is known as exponential decay, that is, exponential functions with a base between 0 and 1), the reciprocal function f(x) = ¹⁄_x are strictly decreasing. f(x) = 3, f(x) = 7 are constant functions.  

Geometrically, a function is increasing or decreasing when, read left to right (as you move from left to right along its graph), the graph is going up (the function rises or remains constant) or down (the function falls or remains constant) respectively. It is constant when its graph is flat.

Increasing/Decreasing Functions

A function is increasing when the graph of the function rises from left to right, i.e., the y-value increases as the x-value increases. More formally, given any x₁, x₂ from an interval I with x₁ < x₂ (∀x₁, x₂ ∈ I, x₁ < x₂), f(x₁) < f(x₂) (figures 1.a., 1.b).

A function is decreasing when the graph of the function decreases from left to right, i.e., the y-value decreases as the x-value increases. More formally, given any x₁, x₂ from an interval I with x₁ < x₂ (∀x₁, x₂ ∈ I, x₁ > x₂), f(x₁) < f(x₂).

The derivative of the function f(x) is used to determine whether a function is increasing, decreasing or constant on an interval.

If f’(x) > 0 for every x on some interval I (∀x ∈ I), then f(x) is increasing on this interval. If f’(x) < 0 for every x on some interval I (∀x ∈ I), then f(x) is decreasing on this interval. If f’(x) = 0 for every x on some interval I (∀x ∈ I), then f(x) is constant on this interval. In other words, the intervals where a function is increasing (or decreasing) correspond to the intervals where its derivative is positive (or negative).

Definition. A critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero. The value of the function at a critical point is called a critical value.

How to find increasing and decreasing intervals

1. Calculate the derivative of the function.
2. Identify Critical Points. As it was previously stated, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.
3. Determine the intervals between these critical points and the endpoints of the function’s domain.
4. Analyze the behavior of f’(x) on these intervals. Given an interval (a, c), choose a value b in this interval, a < b < c. Calculate f’(b). If f’(b) is positive, f(x) is increasing ↗ on (a, c). If f’(b) is negative, f(x) is decreasing on ↘ (a, c).

Solved examples

• f(x) = 3x +5, f’(x) = 3 ⇒ f’(x) > 0 ⇒ f is increasing ↗. All linear functions are either increasing or decreasing over their entire domain. A linear function is a function of the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope determines whether the linear function is increasing or decreasing. If the slope is positive, the line is increasing. If the slope is negative, the line is decreasing.

• f(x) = (x -5)² (Figure i).

f’(x) = 2·(x-5).

Critical points: f’(x) = 0 ⇒ 2(x-5) = 0 ↭ x = 5.

1. x < 5, 2·(x-5) < 0 (or alternatively choose a point, e.g., x = 0, f’(0)=2·(-5)= -10) ⇒ f’(x) < 0 ⇒ f is decreasing ↘
2. x > 5, 2·(x-5) < 0 (or alternatively choose a point, e.g., x = 6, f’(6)=2·1= 2) ⇒ f’(x) > 0 ⇒ f is increasing ↗.

The graph of a quadratic function ax² + bx + c is a parabola. It has an extreme point, called the vertex where the curve changes direction.

If the parabola opens upwards (a > 0), the vertex represents the lowest point on the curve and its y-coordinate is the minimum value of the quadratic function. If the parabola opens downwards (a < 0), the vertex represents the highest point on the curve, and its y-coordinate is the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry $x = \frac{-b}{2a}$, So, the graph of the function is increasing on one side of the axis and decreasing on the other side.

a > 0, Domain(f) = ℝ, Range: y ≥ f($\frac{-b}{2a}$), f is decreasing to the left of x = $\frac{-b}{2a}$ (minimum) and increasing to the right of x = $\frac{-b}{2a}$. a < 0, Domain(f) = ℝ, Range: y ≤ f($\frac{-b}{2a}$), f is increasing to the left of x = $\frac{-b}{2a}$ (maximum) and decreasing to the right of x = $\frac{-b}{2a}$.

• f(x) = $x·e^{-x}$ (Figure ii).

f’(x) = $e^{-x}-xe^{-x} = e^{-x}(1-x).$

Critical points: f’(x) = 0 ⇒[The range of the exponential function is all positive real numbers] 1 -x = 0 ↭ x = 1.

1. x < 1, (1 -x) > 0 ⇒ f’(x) > 0 ⇒ f is increasing ↗.
2. x > 1, (1 -x) < 0 ⇒ f’(x) < 0 ⇒ f is decreasing ↘.

• f(x) = x³ +3x² -9x +12 (Figure iii).

f’(x) = 3x² +6x -9 = 3·(x² +2x -3) = 3·(x +3)(x -1).

Critical points: f’(x) = 0 ⇒ 3·(x +3)(x -1) = 0 ↭ x = -3 or 1, and so there are three intervals to investigate.

1. x < -3, e.g., -4, f’(-4) = 3·(-1)·(-5) = 15 > 0 ⇒ f’(x) > 0 ⇒ f is increasing ↗.
2. -3 < x < 1, e.g., 0, f’(0) = 3·3·(-1) = -9 < 0 ⇒ f’(x) < 0 ⇒ f is decreasing ↘.
3. x > 1, e.g., 2, f’(2) = 3·5·1 = 15 > 0 ⇒ f’(x) > 0 ⇒ f is increasing ↗.

• f(x) = 2x³ +3x² -36x (Figure iv).

f’(x) = 6x² + 6x -36 = 6(x² +x -6) = 6·(x+3)·(x-2).

Critical points: f’(x) = 0 ⇒ 6·(x+3)·(x-2) = 0 ↭ x = -3 or 2.

1. x < -3, e.g., -4, f’(-4) = 6·(-1)·(-6) = 36 > 0 ⇒ f’(x) > 0 ⇒ f is increasing ↗.
2. -3 < x < 2, e.g., 0, f’(0) = 6·3·(-2) < 0 ⇒ f’(x) < 0 ⇒ f is decreasing ↘.
3. x > 2, e.g., 3, f’(3) = 6·6·1 > 0 ⇒ f’(x) > 0 ⇒ f is increasing ↗.

• f(x) = 3x -x³, (Figure 1.e).

f’(x) = 3 -3x² = 3(1-x²) = 3(1-x)(1+x). Critical points x = ±1, f(1)=2, f(-1-)=-2.

1. x < -1, f’(-2) = 3·-3·4 = -9 < 0, f’(x) < 0 ⇒ f is decreasing ↘
2. -1 < x < 1, f’(0) = 3 -3·0 = 3 > 0, f’(x) > 0 ⇒ f is increasing ↗
3. x > 1, f’(2) = 3·-3·4 = -9 < 0, f’(x) < 0 ⇒ f is decreasing ↘

• $\frac{x + 1}{x + 2}$

f’(x) = $\frac{1}{(x+2)^2}$ > 0 ∀ x ∈ ℝ, x ≠ -2 ⇒ f is increasing (-∞, -2) and (-2, ∞).

Critical points are those where the derivate is zero or the derivate is not defined, e.g., in our example, x = -2.

The plot is shown in Figure 1.a.  

• $\frac{x^{3}+4}{x^{2}}$

Another example because you can never have enough of what you don’t really need 😄.

Domain = ℝ - {0}. f is not defined at x = 0.

f’(x) = $\frac{3x^{4}-(x^{3}+4)2x}{x^{4}}= \frac{3x^{4}-2x^{4}-8x}{x^{4}}= \frac{x^{3}-8}{x^{3}}$.

1. f’(x) = 0 ⇒ x = 2. Critical points: 0, 2.
2. x < 0, f’(x) > 0 ⇒ f increasing ↗.
3. 0 < x < 2, f’(x) < 0 ⇒ f decreasing ↘.
4. x > 2, f’(x) > 0 ⇒ f increasing ↗.

The plot is shown in Figure 1.c.

Bibliography

1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn, Andrew Misseldine, blackpenredpen, and MathMajor, YouTube’s channels.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. MIT OpenCourseWare, 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007, YouTube.
8. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.