What is a function? A function is a relation where every \(x\)-value is paired with exactly one \(y\)-value.
Let's take a look at the different representations of functions. Click on the links below to learn more about each representation.
Ordered Pairs
Tables
Graphs
Context
Let's take a look at the different representations of functions. Click on the links below to learn more about each representation.
Ordered Pairs
Tables
Graphs
Context
Function notation: When writing a rule for a function, it should be written using function notation. To write using function notation, change the \( y \) to \( f(x) \). When you see \(f(x) \) it is read as "f of x" or the "function evaluated at x". The \(x\) in the rule still represents the inputs (domain), and \(f(x) \) represents the outputs (range).
For example, the rule \(y = 2x + 1 \) can be written using function notation: \( f(x) = 2x + 1\).
For example, the rule \(y = 2x + 1 \) can be written using function notation: \( f(x) = 2x + 1\).
Evaluating a Function
Using the function example above, \(f(x) = 2x + 1\), the notation \(f(3)\) means that the function is evaluated at \(3\). We do this by substituting \(3\) for \(x\) in the function. So \(f(3) = 2(3) + 1 = 6 + 1 = 7\). Let's look at a couple of examples.
Example 1:
Given the function \(g(x) = -5x - 11\)
a) Find \(g(-8)\)
b) Find \(x\) when \(g(x) = 5\)
Solutions:
a) Substituting \(-8\) for \(x\), \(g(-8) = -5(-8) - 11 = 40 - 11 = 29\).
b) This means the \(y\) value is \(5\). So we solve the equation \(-5x - 11 = 5\)
\(-5x - 11 = 5\)
\(-5x = 16\)
\(x = -\Large\frac{16}{5}\)
Example 2:
What is the value of \(f(7)\) in the graph below?
Using the function example above, \(f(x) = 2x + 1\), the notation \(f(3)\) means that the function is evaluated at \(3\). We do this by substituting \(3\) for \(x\) in the function. So \(f(3) = 2(3) + 1 = 6 + 1 = 7\). Let's look at a couple of examples.
Example 1:
Given the function \(g(x) = -5x - 11\)
a) Find \(g(-8)\)
b) Find \(x\) when \(g(x) = 5\)
Solutions:
a) Substituting \(-8\) for \(x\), \(g(-8) = -5(-8) - 11 = 40 - 11 = 29\).
b) This means the \(y\) value is \(5\). So we solve the equation \(-5x - 11 = 5\)
\(-5x - 11 = 5\)
\(-5x = 16\)
\(x = -\Large\frac{16}{5}\)
Example 2:
What is the value of \(f(7)\) in the graph below?
Solution:
On the graph the point at \(x = 7\) has the coordinates \((7, f(7))\). The value of \(y\) when \(x = 7\) is \(2\) from the graph so \(f(7) = 2\).
On the graph the point at \(x = 7\) has the coordinates \((7, f(7))\). The value of \(y\) when \(x = 7\) is \(2\) from the graph so \(f(7) = 2\).