Hybrid Functions

Hybrid functions, like the name suggests, are functions that are made up of pieces of other functions. We often also call them piecewise functions for this reason.

We will take a look at how to graph a hybrid function and also how to find the equation of one. 

The important thing to note is that each piece of the function has its own domain and range which influence where and how we graph the pieces.

Graphing a hybrid function

Graph the function

We can see that this function is composed of a quadratic function and a linear function.

To begin we can graph the quadratic function, $y=x^2+2$y=x2+2

Using what we know about graphing quadratic functions, we have a minimum turning point at $(0,2)$(0,2).

As we are only graphing the parabola for $x\le1$x≤1 then we need to evaluate this end point.

$y=1^2+2=3$y=12+2=3

So our end point is $(1,3)$(1,3)

We might like at least another point on the curve, so let's use $(-2,6)$(−2,6)

So far we have this piece of the graph drawn.

Now we turn our attention to the second piece, $y=3x$y=3x

The line graph will begin at $x=1$x=1 so we can evaluate this end point and we obtain $(1,3)$(1,3). This means it will connect with the end point of the parabola.

We need two points to draw a straight line so let's use $(3,9)$(3,9).

Adding this to our graph we get the full picture of our hybrid function.

Finding the equation of a hybrid function from the graph

Determine the equation of the hybrid function drawn below.

We can see this time that the function is composed of 3 different functions pieced together.

Working from left to right we can see a linear function, followed by another linear function, and then finally by a quadratic function.

Working with the first linear function (in blue) we can see that the slope is $1$1.

Using any other point we like on the graph, we can calculate the $y$y intercept which is $8$8.

We now have $y=x+8$y=x+8 where $x<-1$x<−1

How did we know to say $x<-1$x<−1 rather than $x\le-1$x≤−1?

We didn't. When we join the pieces of the graph together at the end to define the entire hybrid function, we need a system in place to correctly define the end points. One convention is that when graphing to the left of an $x$x value, we define that portion as less than, rather than less than or equal to.

The second linear graph (in green) is far more straight forward as it's a simple horizontal line.

We have $y=7$y=7 where $-1\le x<2$−1≤x<2

Now let's turn our attention to the final piece of the graph, the parabola.

The parabola connects on to the green linear graph with its turning point (2,7).

It has a maximum turning point and a dilation scale factor of 1.

We therefore have $y=-\left(x-2\right)^2+7$y=−(x−2)2+7 where $x\ge2$x≥2

Putting this all together we have:

Worked Examples:

Question 1

Question 2