Piecewise functions
Finally, functions don't need to be defined by just one equation. A piecewise function specifies different function rules based on certain intervals of $x$x values. These functions may have gaps on their graphs - we call these discontinuous functions - but as long as there is only one $f(x)$f(x) value for any given $x$x value, it is still a function! The image below shows a piecewise function that we can write as:
| $f(x)$f(x) | $=$= |  |
$x^2$x2 | if $x\ge-1$x≥−1 |
| $x+2$x+2 | if $x<-1$x<−1 |
We could use either the graph or the equations to evaluate function values, but let's practice using the equations by choosing an $x$x value that isn't shown in the sketch.
For example, let's find $f(5)$f(5). This requires an $x$x value greater than $-1$−1, so we choose the top equation in the piecewise set. So, $f(5)=x^2=25$f(5)=x2=25.
Practice Questions
Question 1
Consider the function graphed below.
What is the value of the function at $x=-1$x=−1?
Is the graph continuous or discontinuous at $x=-1$x=−1?
Continuous
ADiscontinuous
BWhat is the value of the function at $x=0$x=0?
Is the graph continuous or discontinuous at $x=0$x=0?
Continuous
ADiscontinuous
BWhat is the value of the function at $x=2$x=2?
Is the graph continuous or discontinuous at $x=2$x=2?
Continuous
ADiscontinuous
B
Question 2
A function is defined as
| $f(x)$f(x) | $=$= | |
$x^2-2$x2−2 | if $x\ge3$x≥3 |
| $4$4 | if $0 |
|||
| $4x$4x | if $x\le0$x≤0 |
Evaluate $f\left(4\right)$f(4).
Evaluate $f\left(0.5\right)$f(0.5).
Evaluate $f\left(-3\right)$f(−3).
Evaluate $f\left(a\right)$f(a) if $a$a is a negative value.