April 13, 2024

Types of functions

Functions are mathematical relationships where each input has a single output. You have probably been doing functions since you began learning maths, but they may have looked like this:

Types of functions

Algebraically this is:

f (x) = x + 10,

here x =16, y =26.

We can use graphs to show multiple outputs of y for inputs x, and therefore visualize the relation between the two. Two common types of functions are linear functions and quadratic functions.

2.1.1 Linear functions

Linear functions y = m x + c increases/decreases at a constant rate m, where m is the gradient and c is the y-intercept

Linear functions 1
Linear functions 2

Straight line equations are sometimes written in two other forms, which you should be comfortable rearranging them to and from:

ax + by + d  = 0  ⇒ general form
y − y1 = m(x − x1) ⇒ point-slope form

Example: Determine the midpoint, length and gradient of the straight line connecting the two points P1(2, 8) and P2(6, 3)

Linear functions 3
Linear functions 4

2.1.2 Quadratic functions

Graphed, quadratic functions make a parabolic shape; they increase/decrease at an increasing rate.

Quadratic functions 2
Quadratic functions 3

Quadratic functions y = ax2 + bx + c = 0

Axis of symmetry x = −b/2a = x−coordinate of vertex

Roots the x-intercept(s). Algebraically, roots can be found through factorisation or using the quadratic formula

Quadratic formula used to find the roots if a ≠ 1

Quadratic formula

Discriminant ∆ the b2 − 4ac part of the formula, which can be used to determine how many x-intercepts a quadratic equation has

     ∆ > 0   ⇒  2roots

     ∆ = 0   ⇒  1root

     ∆ < 0   ⇒  no real roots

Note: Roots of quadratic equations are also referred to as ‘solutions’ or ‘zeros’.

Two more standard ways to write quadratic equations are:

  y = a(x − h)2 + k      with the vertex at (h, k)
  y = a(x − p)(x − q)    with x-intercepts (p, 0)(q, 0)

Solving quadratic equations by factorisation

   Solve x− 5x + 6 = 0

 

  1. Set up a system of equations  p + q = b and p × q = c

 

Solving quadratic equations by factorisation

2. Plug the values for p and q into

        (x + p)(x + q)

(x − 2)(x − 3) = x2 − 5x + 6

3. Equate each part to 0

    (x + p) = 0, (x + q) = 0,

    and solve for x

Solving quadratic equations by factorisation 2

Solving quadratic equations using the quadratic formula

   Solve 3x2 − 8x + 4 = 0

  1. Calculate the discriminant ∆ = b2 − 4ac

∆ = (−8)2 − 4 · 3 · 4 = 16

2. Use the discriminant to determine the number of solutions

∆ > 0 so 2 solutions

3. Find solutions using quadratic formula

Solving quadratic equations using the quadratic formula 1
Solving quadratic equations using the quadratic formula 3

Through a method called completing the square, you can rearrange a quadratic function into the form y = a(x − h)2 + k. This way you can find the coordinates of the vertex (the minimum or maximum). For the exam you will always be asked explicitly to use this method.

Find the vertex by completing the square

     Express f(x) = 4x2 − 2x − 5 = 0 in the form y = a(x − h)2 + k. Hence, find the coordinate of the vertex of f(x).
  1. Move c to the other side
4x2 − 2x = 5

2. Divide by a

Find the vertex by completing the square 2
3. Calculate (x coeficient/2) 2
Find the vertex by completing the square 3

4. Add this term to both sides

Find the vertex by completing the square 4

5. Factor perfect square and bring constant back

Find the vertex by completing the square 5

2.1.3  Functions with asymptotes

Asymptote a straight line that a curve approaches, but never touches.

A single function can have multiple asymptotes: horizontal, vertical and in rare cases diagonal. Functions that contain the variable (x) in the denominator of a fraction and exponential and logarithmic functions will always have asymptotes.

Vertical asymptotes

Vertical asymptotes occur when the denominator is zero, as dividing by zero is undefinable. Therefore if the denominator contains x and there is a value for x for which the denominator will be 0, we get a vertical asymptote.

Example: In the function f (x) = x/x − 4 the denominator is 0 when x = 4, so this line forms the a vertical asymptote.

Horizontal asymptotes

Horizontal asymptotes are the value that a function tends to as x becomes really big or really small; technically speaking to the limit of infinity, x → ∞. The general idea is then that when x is large, other parts of the function not involving x become insignificant and so can be ignored.

Example: 

Horizontal asymptotes 4

Exponential and logarithmic functions

Exponential functions will always have a horizontal asymptote and logarithmic functions will always have a vertical asymptote, due to the nature of these functions. The position of the asymptote is determined by constants in the function.

Exponential

f(x) = ax + c     asymptote at y = c

where a is a positive number (often e)

Exponential

Logarithmic

g(x) = loga (x + b)

asymptote at x = −b

Logarithmic

2.1.4 Special Functions

               The function (ax + b)/(cx +d)

Rational function of the form y = (ax + b)/(cx +d)

The graph made by this function has one horizontal and one vertical asymptote. The graph is not continuous in all points, but splits into two parts. Both of the parts approach horizontal asymptote at either negative or positive large values of x and approach horizontal asymptote at either negative or positive large values of y. Also both parts are located in different “corners” of the coordinate system.

The vertical asymptote occurs where denominator is equal to 0, meaning x = −d/c . The graph is not defined at that point.

The horizontal asymptote occurs at very large values of x, meaning that asymptote occurs at

Special Functions 2
Special Functions 3