Topic 2 - Functions and Equations

Introduction

The general aim of this section is to explore the notion of function as a unifying theme. Additionally, candidates should be able to apply functional methods to a variety of situations. Use of a GDC (Graphing Display Calculator) is expected.

Concept of a Function

Composite Functions

A composite function is a function made up of multiple parts, or steps. Before encountering composite functions, you will have seen functions of the form f(x), g(x) and so on. A composite function has another function inside it, and could for example be written as f(g(x)), or g(f(x)). These are said, "f of g of x" and "g of f of x". f(g(x)) means that the function 'g' is applied to 'x', and then following this, the function 'f' is applied to the output of the function 'g'. For example, if g(x)=2x+3, and f(x)=x^2, and the composite function was f(g(x)), 'x' would have 'g' applied, and become '2x+3'. '2x+3' subsequently has 'f' applied, and becomes (2x+3)^2.

f(g(x)) does NOT equal g(f(x)).

Inverse Function

The inverse function, is as its name signifies, the inverse of a function, shown as ${\displaystyle f}$ ${\displaystyle -1}$ ${\displaystyle (x)}$  (f(x)^-1). This is accomplished by substituting ${\displaystyle x}$  and ${\displaystyle y}$  for one another within the equation, and evaluating the function to where you aim to get the ${\displaystyle y}$  variable alone, again.

Examples:

Ex.1

${\displaystyle f(x)=x^{2}\,\!}$

${\displaystyle y=x^{2}\,\!}$

${\displaystyle (x)=(y)^{2}\,\!}$

${\displaystyle {\sqrt {x}}=y\,\!}$

${\displaystyle f\,\!}$ ${\displaystyle -1}$ ${\displaystyle (x)={\sqrt {x}}\,\!}$

Ex.2

${\displaystyle f(x)=3x-2\,\!}$

${\displaystyle y=3x-2\,\!}$

${\displaystyle (x)=3(y)-2\,\!}$

${\displaystyle x+2=3y\,\!}$

${\displaystyle {\frac {x+2}{3}}=y\,\!}$

${\displaystyle f\,\!}$ ${\displaystyle -1}$ ${\displaystyle (x)={\frac {x+2}{3}}\,\!}$

The Graph of a Function

Horizontal and Vertical Asymptotes

An asymptote can be described as a line that represents the end behavior of a function. While they may be crossed, they may not be crossed at an infinite number of points. They can be horizontal, vertical, or oblique (diagonal).

For instance, if you look at a visual representation of ${\displaystyle y=1/x}$  you will see that while the graph approaches the x-axis, the line ${\displaystyle y=0}$  it will never touch the line.

Exponents of the Variables

When dealing with a function, it s always a good idea to take a look at the highest power the variable(s) are to, among other things. For example, for the equations of ${\displaystyle y=x}$  and ${\displaystyle y=x^{2}}$  the behaviors of these functions differ greatly. With ${\displaystyle y=x}$  the function extends from negative infinity from Quadrant III to positive infinity in Quadrant I. While with the function of ${\displaystyle y=x^{2}}$  the function extends from Quadrant II to Quadrant I.

The Reciprocal Function

X ----> 1/x, i.e. f(x) = 1/x is defined as the reciprocal function.


Notice that:

- f(x) = 1/x is meaningless when x = 0 - the graph of f(x) = 1/x exists in the first and third quadrants only - f(x) = 1/x is symmetric about y = x and y = -x - f(x) = 1/x is asymptotic (approaches) to the x-axis and to the y-axis

(Source: Mathematics for the international student, Mathematics SL, International Baccalaureate Diploma Programme by John Owen, Robert Haese, Sandra Haese, Mark Bruce)

Standard Form

${\displaystyle y=ax^{2}+bx+c\,\!}$

Vertex or Turning Point Form

${\displaystyle y=a(x-h)^{2}+k\,\!}$  , where (h,k) is the vertex

${\displaystyle (h,k)=({\frac {-b}{2a}},{\frac {4ac-b^{2}}{4a}})}$

-b/(2a)

Roots of the equation

${\displaystyle {\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$

Where b2-4ac is the discriminant. It can also be written as Δ.

When Δ>0, the equation has 2 distinct, real roots.

When Δ=0, the equation has 2 repeated real roots.

When Δ<0, the equation has no real roots (only imaginary roots).

Exponential Function

In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The exponential function is also often written as exp(x), especially when x is an expression complicated enough to make typesetting it as an exponent unwieldy.

The graph of y = ex is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some older sources refer to the exponential function as the anti-logarithm.

Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e.