Functions

Basics of Functions

Function: a mathematical relationship where each input has a single output. It is often written as $f(x)$ where $x$ is the input.

Domain: all possible $x$ values, the input.

Range: all possible $y$ values, the output.

Types of Basic Functions

Linear Functions:

  • General Form: $y = mx+c$
  • Increases/decreases at a constant rate $m$, where $m$ is the gradient and $c$ is the y-intercept.
  • Gradient: $m = \frac{y_2-y_1}{x_2-x_1}$
  • Parallel lines: $m_1 = m_2$
  • Perpendicular lines: $m_1 = -\frac{1}{m_2}$

The equations for linear functions are written in three different forms:

  • Gradient form: $y=mx+c$
  • General form: $ax+by+d=0$
  • Point-slope form: $y-y_1=m(x-x_1)$

Quadratic Functions:

  • General Form: $y = ax^2+bx+c$
  • Axis of symmetry: $x=-\frac{b}{2a}$, which is also the x-coordinate of the vertex
  • Roots: the x-intercept(s). Algebraically, roots can be found through factorisation or using the quadratic formula
  • Quadratic Formula: $\frac{-b±\sqrt{b^2-4ac}}{2a}=\frac{-b\pm\sqrt{Δ}}{2a}$
  • Discriminant (Δ): $b^2 - 4ac$ part of the formula, which can be used to determine how many x-intercepts a quadratic equation has
    • If Δ > 0 ⇒ 2 roots
    • If Δ = 0 ⇒ 1 root
    • If Δ < 0 ⇒ no real roots

The equations for quadratic functions can be written in three different ways:

  • General form: $y=ax^2+bx+c$
  • Vertex form: $y=a(x-h)^2+k$, with the vertex at $(h,k)$
  • Factored form: $y=a(x-r)(x-s)$, with the x-intercepts being $(r,0)$ and $(s,0)$

Through a method called completing the square, you can rearrange a general 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).


  • Step 0: Start with the equation $y=ax^2+bx+c$
  • Step 1: Divide all terms by $a$ (the coefficient of $x^2$).
  • Step 2: Move the constant term $\left(\frac ca\right)$ to the left side of the equation.
  • Step 3: Complete the square on the right side of the equation (add $\left({\frac{b}{2a}}\right)^2$, then factor) and balance this by adding the same value to the left side of the equation.
  • Step 4: Subtract the number that remains on the left side of the equation, then multiply everything by $a$, to get the vertex form.

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, as well as 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.


Horizontal asymptotes: Horizontal asymptotes are the value that a function tends to as $x$ becomes really big or really small; technically 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.


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 function: $f(x)=a^x+c$, where there is an asymptote at $y=c$
  • Logarithmic function: $g(x)=\log_a(x+b)$, where there is an asymptote at $x=-b$

Exponents and Logarithms

Exponent Laws:


Exponents always follow certain rules. If you are multiplying or dividing, use the following rules to determine what happens with the powers:

  • $a^m \cdot a^n = a^{m+n}$
  • $\frac{a^m}{a^n} = a^{m-n}$
  • $(a^m)^n = a^{mn}$
  • $a^0 = 1$
  • $a^{-n} = \frac{1}{a^n}$
  • $a^{\frac{x}{y}} = \sqrt[y]{a^x}$

Logarithm Laws:


Logarithms are the inverse mathematical operation of exponents, like division is the inverse mathematical operation of multiplication. The logarithm is often used to find the variable in an exponent.


$a^x=b \iff x=\log_ab$


Logarithms with bases of 10 and e have special notations in which their base is not explicitly noted:

  • Common logarithm: $\log_{10}x = \log x$
  • Natural logarithm: $\log_{e}x = \ln x \ \ \ \ $ (this will be discussed later in greater detail)

The following are important laws of logarithms that will be useful when simplifying or solving equations:

  • $\log_a(xy) = \log_a x + \log_a y$
  • $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
  • $\log_a(x^n) = n \cdot \log_a x$
  • $\log_a a = 1$
  • $\log_a 1 = 0$
  • $\log_a b = \frac{\log_c b}{\log_c a}$

Rational Functions

General form: $\ y=\frac{ax+b}{cx+d}$


The graph made by these functions has one horizontal and one vertical asymptote. The vertical asymptote occurs where the denominator is equal to 0, or when $x=-\frac dc$. The horizontal asymptote occurs at very large values of $x$: Hence, since $y=\frac{ax+b}{cx+d}$, as $x→∞$, $y→\frac ac$.

Inverse Functions

Inverse functions are the reverse of a function. Essentially, you are switching the $x$ and $y$ values (domain and ranged are swapped). The inverse function of $f(x)$ is notated as $f^{-1}(x)$, where $f^{-1}(f(x))=x$

Composite Functions

Composite functions are a combination of two functions. They are usually written as: $(f∘g)(x)$ or $f(g(x))$, which is read as "f of g of x". To find the value of the composite function above for some $x$, first evaluate $f(x)$, then input that number into $g(x)$ to get $f(g(x))$.

Transforming Functions

Functions can be transformed by adding, subtracting, multiplying, or dividing constants at different parts of the function. This results in a change in behaviour of the function, and a modified version of the parent function's graph.


Common transformations applied to $f(x)$:

  • Vertical stretch by a factor of $a$ $\implies$ $a⋅f(x)$
  • Horizontal stretch by a factor of $\frac 1a$ $\implies$ $f(a⋅x)$
  • Reflection in the x-axis $\implies$ $-f(x)$
  • Reflection in the y-axis $\implies$ $f(-x)$
  • Vertical shift upwards by $a$ units $\implies$ $f(x)+a$
  • Horizontal shift to the left by $a$ units $\implies$ $f(x+a)$