Function Fundamentals

Definition. A function f is a rule, relationship, or correspondence that assigns to each element of one set (x ∈ D), called the domain, exactly one element of a second set, called the range (f(x) ∈ E). Typically, the sets D and E will be both the set of real numbers, ℝ.

Let f be a function, a, b ∈ Domain(f). If a = b, then f(a) = f(b), e.g., f: ℚ → ℚ, f(p/q) = -p is not a function because f(¹⁄₂)=f(²⁄₄), but ¹⁄₂ = ²⁄₄.

The domain of a function is the set of all inputs, it consists of all the elements for which f is defined on. The range of a function is the set of all outputs.

There are many was to represent a function:

• Verbally. It is a means of describing the relationship between inputs and outputs with words and sentences, e.g., “y is double the value of x minus five,” and “six added to two times x”.
• Numerical. The function is expressed as a list of value pairs, {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}, {(1, 1), (4, 2), (9, 3), (16, 4), (25, 5)}

  x	1	2	3	4	5	6	7
  f(x)	2	4	6	8	10	12	14

• A function can be represented algebraically, by giving an algebraic expression or formula, e.g., 2x + 5, $\sqrt{x}, e^x, x^2-2x +4, etc.$

If a domain is not stated explicitly, the domain is the set of all number for which the expression makes sense and defines a real number, e.g., f(x) = x³ + 2x² + 4, Domain(f) = ℝ; g(x) = $\sqrt{x-2}$, Domain(g) = ℝ - {x ≥ 2} = [2, +∞); h(x) = $\frac{2x+3}{x^2-x}+4,$ Domain(h) = (-∞, 0) ∪ (0, 1) ∪ (1, ∞). r(x) = $\frac{\sqrt{x}}{x^2 - x},$ Domain(r) = (0, 1) ∪ (1, ∞) = {x ∈ ℝ: 0 < x < 1 or x > 1}.

• It can also be represented graphically or visually. We plot a series of inputs and outputs and connect them with a smooth line (Figure 1.a.)

 

Evaluating a function means finding the value of f(x) that corresponds to a given value of x Examples:

• f(x) = $x^2-2x +4$, f(2) = 4 - 4 + 4 = 0, f(0) = 4, f(1) = 1 -2 +4 = 3.

• x	1	2	3	4	5	6	7
  f(x)	2	4	6	8	10	12	14

1. Domain(f) = {1, 2, 3, 4, 5, 6, 7}. Range(f) = {2, 4, 6, 8, 10, 12, 14}
2. f(5) = 10, f(2) = 4, f(3) = 6, but f(9) is undefined.
3. Solve the equation f(x) = 12. The solution is x = 6. Solve the equation f(x) = 5. The solution does not exist.
4. Solve the inequality f(x) > 10. The solution is x = 6 or 7.

 

The Vertical Line Test

The vertical line test is a graphical method of determining whether a curve represents the graph of a function or not by visually examining the number of intersections of the curve with vertical lines. A vertical line needs to intersect the graph of a function at most at one point for it to represent a function.

Example: f(x) = x + 2, y = x³ are functions, but y² = x ↭ y = $±\sqrt{x}$ is not a function (e.g., x = 4, y = ±2).

 

Intercepts of Graphs

Definition. The x-intercept of a graph is the x-coordinate of any point on the graph that intersects the x-axis. Equivalently, the y-intercept is the y-coordinate of any point on the graph that intersects the y-axis.

Example. Let’s find the x- and y-intersect of -2x + 4y = 8. To find the x-intercept, set y = 0 ⇒ -2x = 8 ⇒ x = $\frac{8}{-2}=-4$. To find the y-intercept, set x = 0 ⇒ 4y = 8 ⇒ y = 2 (Figure 1). Not all graphs necessarily have both intercepts: y = 3 has a y-intercept of (0, 3) and no x-intercept, but x = 1 has a x-intercept of (1, 0) and no y-intercept (Figure 2 and 3 respectively).

The graph y = x² -5x + 6 intersects the x-axis in two places, namely (2, 0) and (3, 0). Its only y-intercept is (0, 5) -Figure 4-. Figure 5 and 6 illustrate that even though those graphs do not determine functions, they still have x- and y-intercepts (the circle do not have y-intercept).

Monotonic functions

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. 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

Geometrically, a function is increasing or decreasing when, read left to right, the graph is going up or down respectively. It is constant when its graph is flat.

A function is concave up or convex if it bends upwards, the line segment or chord between any two points A₁, A₂, on the graph of the function lies above the graph between the two points (the midpoint B lies above the corresponding point A₀ of the graph of the function). f is concave down if -f is convex (it bends downwards).

Formally, f is convex on an interval [a, b] if ∀ x₁, x₂ ∈ [a, b], 0 < θ <1, the following inequality holds: f(θa + (1-θ)b) ≤ θf(a) + (1-θ)f(b) or

The function f (Figure 1) is increasing on (-6, -5) ∪ (-3, 0) and decreasing on (-5, -3) ∪ (4, 7). It is constant on [0, 4]. Futhermore, f is concave upward on (-4, -2) ∪ (4, 7) and concave downward on (-6, -4) ∪ (-2, 0) ∪ (4, 7).

 

Local Extrema

Let D be the domain of a function and c ∈ D. f is said to have a local or relative maximum point at c if there exists an interval (a, b) containing c such that f(c) is the maximum value of f on (a, b) ∩ D: f(c) ≥ f(x) ∀ x in (a, b) ∩ D. f is said to have a local or relative minimum point at c if there exists an interval (a, b) containing c such that f(c) is the minimum value of f on (a, b) ∩ D: f(c) ≤ f(x) ∀ x in (a, b) ∩ D.

A function has a local or relative extremum at c if c is either a relative maximum or minimum, that is, a point of the graph where f changes its monotonicity within the domain. Futhermore, a point of inflection is a point where the graph of f changes its concavity.

In the previous example, f has local or relative minimum points at -6, -3, and 7, and local or relative maximum points at -5, 0, and 4. On the open interval (0, 3), the function is constant, and therefore every point is both a local or relative maximum and minimum point. Futhermore, (-4, 0), (-2, 0), (0, 6), and (4, 6) are points of inflection.

Absolute Extrema

The maxima and minima (the respective plurals of maximum and minimum) of a function are the largest and smallest value of the function, either within a given range (the local or relative maximum and minimum; you can visualize them as kind of local hills and valleys), or on the entire domain (the global or absolute maximum and minimum).

Formally, a function f defined on a domain D has a global or absolute maximum point at c if f(c) ≥ f(x) ∀ x in D. Similarly, the function f has a global or absolute minimum point at c if f(c) ≤ f(x) ∀ x in D.

Figure 3 illustrates a function that does not have an absolute maximum. Besides, a graph can only have one absolute minimum or maximum, but multiples x-values could obtain them as it is illustrated on figure 4. It has an absolute maximum value of y = 3.25 which is obtained at x = 0.775 and 3.225.

 

Symmetry of Graphs

Definition. A function f(x) is even if f(-x) = f(x) for all the values of x. A function f(x) is odd if f(-x) = -f(x). An even function has reflection symmetry about the y-axis. Even functions allow us to view the y-axis as a mirror, they are symmetric with respect to the y-axis, e.g., x², |x|, and cosx. Otherwise, an odd function has symmetry about the origin. Some examples are x³, sin(x), and ¹⁄_x.

Let f(x) = 3x -x³. f(-x) =-3x - (-x)³ = -3x +x³ = - (3x -x³) = -f(x), so f is an even function, it has symmetry about the origin.

 

Algebra of functions

We can combine existing functions. One way is to use function composition, another one is to carry out the basic four arithmetic operations on functions. For any arithmetic operation of two functions at an input, we just have to apply the same operation with the function outputs.

Let f and g be functions:

• (f + g)(x) = f(x) + g(x), e.g., f(x) = 3x, g(x) = -x³, (f + g)(x) = 3x -x³

  For doing any arithmetic operation of two functions, their domains must be the same (it is typically the set of all real numbers, ℝ) or the domain of the sum is the intersection of the domain of both functions, i.e., Domain(f + g) = Domain(f) ∩ Domain(g), Domain(f - g) = Domain(f) ∩ Domain(g), and Domain(fg) = Domain(f) ∩ Domain(g).

• (f - g)(x) = f(x) - g(x), e.g., f(x) = x², g(x) = x -1, (f - g)(x) = x² -x +1.
• (fg)(x) = f(x)g(x), e.g., f(x) = x, g(x) = 2, (fg)(x) = 2x.
• (f/g)(x) = f(x)/g(x), Domain(f/g) = Domain(f) ∩ Domain(g) ∩ {x | g(x) ≠ 0} We need to take extra care of an additional condition so the denominator is not equal to zero because division by zero is undefined. Example: f(x) = x, g(x) = x -1, $(\frac{f}{g})(x)=\frac{x}{x-1}, ~Domain(\frac{f}{g})=ℝ$ - {x| g(x) ≠ 0} = ℝ - {1}.

Inverse Functions

The inverse function of a function f is a function that undoes the operation of f.For a function f, its inverse g(=f^-1), g(f(x))=x. An example is f(x)=$\sqrt{x},$ g(f(x))=x, or g(x)= x², -1.a.- In other words, the square function (x²) is the inverse of the square root function ($\sqrt{x}$).

If you want to plot the graph of f^-1, you need to reflect the graph of f(x) about the line y = x.  

Another example is tanx. Arctangent is the inverse of the tangent, which is the ratio of the opposite side to the adjacent side in a right triangle.

Exponential and logarithms functions.

An exponential function is a function of the form f(x) = a^x, where “x” is a variable and “a” is a constant which is called the base of the function and it should be a positive real number (a>0). Examples are 2^x, 7^x, and (¹⁄₄)^x. There are three kinds of exponential functions depending on whether a > 1, a = 1, 0 < a < 1 -1.a.-  

Properties:

1. The domain of an exponential function is the set of all real numbers.
2. It is always positive, a^x > 0, ∀x ∈ ℝ
3. Exponent rules. If a, b > 0 and x, y are real numbers, then: a^xa^y=a^x+y, $\frac{a^{x}}{a^{y}}=a^{x-y}$, $(a^{x})^{y}=a^{xy}=(a^{y})^{x}$, $a^{x}b^{x}=(ab)^{x}$.

We call the inverse of a^x the logarithmic function with base a, that is, log_ax=y ↔ a^y=x. The logarithm with base e is the natural logarithm. The natural logarithm is the inverse of e^x, that is, lnx=y ↔ a^y=x, -1.b.-

Some properties are:

1. log_a(AB)=log_aA + log_aB
2. log_a(^A⁄_B) = log_aA - log_aB
3. log_a(Aⁿ) = nlog_aA
4. ln1 = 0, lne = 1.