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Functions and Relations

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An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input.

What is a Function?

Some relationships make sense and others don’t. Functions are relationships that make sense. All functions are relations, but not all relations are functions.


A function is a relation that for each input, there is only one output.

Here are mappings of functions. The domain is the input or the x-value, and the range is the output, or the y-value.

realation one

Each x-value is related to only one y-value.

realation two

Athough the inputs equal to -1 and 1 have the same output, this relation is still a function because each input has just one output.

This mapping is not a function. The input for -2 has more than one output.

realation three

Graphing Functions

Using inputs and outputs listed in tables, maps, and lists, makes it is easy to plot points on a coordinate grid. Using a graph of the data points, you can determine if a relation is a function by using the vertical line test. If you can draw a vertical line through a graph and touch only one point, the relation is a function.

Take a look at the graph of this relation map. If you were to draw a vertical line through each of the points on the graph, each line would touch at only one point, so this relation is a function.

realation four

realation in koordinates

Special Functions

Special functions and their equations have recognizable characteristics.

Constant Function

$f(x) = c$

The c-value can be any number, so the graph of a constant function is a horizontal line. Here is the graph of $f(x) = 4$

rfunction one

Identity Function

$f(x) = x$

For the identity function, the x-value is the same as the y-value. The graph is a diagonal line going through the origin.

rfunction two

Linear Function

$f(x) = mx + b$

An equation written in the slope-intercept form is the equation of a linear function, and the graph of the function is a straight line.

Here is the graph of $f(x)= 3x +4$

rfunction three

Absolute Value Function

$f(x) = |x|$

The absolute value function is easy to recognize with its V-shaped graph. The graph is in two pieces and is one of the piecewise functions.

function four

This is just a sample of the most common special functions.

Inverse Functions

An inverse function reverses the inputs with its outputs.

$f(x) = 3x - 4$

Change the inputs with the outputs to create the inverse of this function.

$ \begin{array}{rccl} f(x) &=& 3x -4\\ y &=& 3x -4\\ x &=& 3y -4\\ x +4 &=& 3y -4 + 4\\ x+ 4&=& 3y\\ \frac{x + 4}{3}&=& \frac{3}{3}y\\ f^{-1}(x)&=&\frac{x + 4}{3} \end{array} $

The inverse of $f(x) = 3x - 4$ is $f^{-1}(x) =\frac{x + 4}{3}$.

Not every inverse of a function is a function, so use the vertical line test to check.

Function Operations

You can add, subtract, mutiply, and divide functions.

  • $f(x) + g(x) = (f + g)(x)$
  • $f(x) - g(x) = (f - g)(x)$
  • $f(x) \times g(x) = (f \times g)(x)$
  • $\frac{f(x)}{g(x)}= \frac{f}{g}(x)$

Look at two examples of function operations:

What is the sum of these two functions? Simply add the expressions.

  • $f(x) = 2x + 3$
  • $g(x) = 3x + 5$
  • $(f + g) (x) = 2x + 3 + 3x + 5 = 5x + 8$

What is the product of these two functions? Simply multiply the expressions.

$\begin{array}{rccl} f(x) &=& x + 4\\ g(x) &=& x + 7\\ (f\times g)(x) &=& (x + 4) \times (x +7) &= x^{2} + 11x + 28 \end{array}$