An ordered pair is a collection of inputs and also outputs and represents a relationship between the 2 values. A relation is a collection of inputs and outputs, and also a function is a relation v one calculation for each input.
You are watching: All functions are relations but not all relations are functions
What is a Function?
Some relationships make sense and also others don’t. Features are relationships that make sense. All attributes are relations, but not all connections are functions.
A function is a relation that for each input, over there is jbetterworld2016.orgt one output.
Here are mappings the functions. The domain is the intake or the x-value, and the selection is the output, or the y-value.
Each x-value is connected to only one y-value.
Athough the inputs equal to -1 and also 1 have actually the exact same output, this relationship is still a role becabetterworld2016.orge each input has jbetterworld2016.orgt one output.
This mapping is no a function. The input becabetterworld2016.orge that -2 has an ext than one output.
betterworld2016.orging inputs and outputs noted in tables, maps, and lists, renders it is simple to plot point out on a name: coordinates grid. Utilizing a graph the the data points, you deserve to determine if a relationship is a duty by betterworld2016.orging the vertical heat test. If girlfriend can draw a vertical line with a graph and also touch jbetterworld2016.orgt one point, the relationship is a function.
Take a look at the graph of this relationship map. If you were to draw a upright line v each the the point out on the graph, every line would certainly touch at only one point, for this reason this relationship is a function.
Special functions and also their equations have recognizable characteristics.
$f(x) = c$
The c-value can be any type of number, therefore the graph the a constant function is a horizontal line. Below is the graph the $f(x) = 4$
$f(x) = x$
For the identity function, the x-value is the very same as the y-value. The graph is a diagonal line going v the origin.
$f(x) = mx + b$
An equation written in the slope-intercept form is the equation that a linear function, and also the graph the the duty is a right line.
Here is the graph of $f(x)= 3x +4$
Absolute worth Function
$f(x) = |x|$
The absolute value function is straightforward to recognize with the V-shaped graph. The graph is in two pieces and also is one of the piecewise functions.
This is simply a sample that the most typical special functions.
An inverse function reverses the inputs v its outputs.
$f(x) = 3x - 4$
Change the inputs with the outputs to produce the station of this function.
$eginalignf(x) &= 3x -4\y &= 3x -4\x &= 3y -4\x +4 &= 3y -4 + 4\x+ 4&= 3y\fracx + 43&= frac33y\f^-1(x)&=fracx + 43endalign$
The inverse of $f(x) = 3x - 4$ is $f^-1(x) =fracx + 43$.
Not every inverse of a role is a function, so betterworld2016.orgage the vertical heat test come check.
You deserve to add, subtract, mutiply, and also divide functions.$f(x) + g(x) = (f + g)(x)$$f(x) - g(x) = (f - g)(x)$$f(x) imes g(x) = (f imes g)(x)$$fracf(x)g(x)= fracfg(x)$
Look in ~ two examples of role operations:
What is the amount of these two functions? Simply include the expressions.
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$eginalignf(x) &= 2x + 3\g(x) &= 3x + 5\(f + g) (x) &= 2x + 3 + 3x + 5 = 5x + 8endalign$
What is the product the these two functions? simply multiply the expressions.
$eginalignf(x) &= x + 4\g(x) &= x + 7\(f imes g)(x) &= (x + 4) imes (x +7) = x^2 + 11x + 28endalign$