Consider the absolute value function definition given below and the associated graph.
$\left|3x+2\right|$|3x+2|$=$= | $3x+2$3x+2$,$, | $x\ge-\frac{2}{3}$x≥−23 | |
$-\left(3x+2\right)$−(3x+2)$,$, | $x<-\frac{2}{3}$x<−23 |
The domain of this function is the whole real line - any value of $x$x can be substituted in, large or small, positive or negative. On the other hand, the range of the absolute value function illustrated above is the interval $[0,\infty)$[0,∞). Any non-negative value is a possible output, but we cannot obtain a negative output from any $x$x-value.
Also note that $|-3x-2|$|−3x−2| is just another way to define the same function. For a given value of $x$x the function $-3x-2$−3x−2 (without absolute value signs) has almost the same output as the function $3x+2$3x+2 - it just differs by its sign. The absolute value around each of these functions then makes the outputs exactly the same.
This is true in general: For all values of $a$a and $b$b (each positive or negative),
$|-ax-b|=|(-1)\times(ax+b)|=|-1|\times|ax+b|=|ax+b|$|−ax−b|=|(−1)×(ax+b)|=|−1|×|ax+b|=|ax+b|.
This trick can be convenient whenever we want to move minus signs around inside the absolute value.
Solve: Describe the natural domain and the range of the function given by the expression $|-x+2|$|−x+2|.
Think: The absolute value function can take any number into it - no values of $x$x can break the function. What kinds of numbers $k$k could be the output? In mathematical terms, for what values of $k$k can $k=|-x+2|$k=|−x+2|? Think about what value of $x$x makes $k=0$k=0. We could also use the trick discussed above and discuss the expression $|x-2|$|x−2|, since it is equal to $|-x+2|$|−x+2|, but it is not necessary.
Do: The domain of the function is the entire real line, $\left(-\infty,\infty\right)$(−∞,∞).
As for the range, we can tell that $x=2$x=2 produces $k=0$k=0, and any value of $x$x either less than or greater than $2$2 will produce a positive value of $k$k. To confirm this, let's consider each case separately.
If $x>2$x>2 then $-x+2$−x+2 will be negative. This means
$k=-(-x+2)=x-2$k=−(−x+2)=x−2,
and by substituting a large enough $x$x we can make $k>0$k>0 as large as we like.
If $x<2$x<2 then $-x+2$−x+2 will be positive. This means $k=-x+2$k=−x+2, and by substituting a large enough $x$x we can make $k>0$k>0 as large as we like once again.
Taking these cases together we conclude that $k\ge0$k≥0 is the range. We have confirmed in algebraic symbols what is evident from a graph of the function: the range is the non-negative real numbers.
Reflect: Do all absolute value functions have domain $\left(-\infty,\infty\right)$(−∞,∞) and range $\left[0,\infty\right)$[0,∞)? When might this not be the case?
Consider the function that has been graphed.
What is the domain of the function?
all real $x$x
$x<3$x<3
$x\ge0$x≥0
$x>0$x>0
all real $x$x
$x<3$x<3
$x\ge0$x≥0
$x>0$x>0
What is the range of the function? Give your answer as an inequality.
Consider the graph of the function below.
What is the domain of the function?
$\left(0,\infty\right)$(0,∞)
$\left(-\infty,2\right)$(−∞,2)
$\left[0,\infty\right]$[0,∞]
$\left(-\infty,\infty\right)$(−∞,∞)
$\left(0,\infty\right)$(0,∞)
$\left(-\infty,2\right)$(−∞,2)
$\left[0,\infty\right]$[0,∞]
$\left(-\infty,\infty\right)$(−∞,∞)
What is the range of the function? Give your answer using interval notation.
Consider the function defined as $y=\left|4x+12\right|$y=|4x+12|.
Fill in the gaps to completely define the expression $\left|4x+12\right|$|4x+12|.
$-4x-12$−4x−12 | for $x$x$<$<$\editable{}$ | |||||
$\left|4x+12\right|$|4x+12| | $=$= | $\editable{}$ | for $x=-3$x=−3 | |||
$4x+12$4x+12 | for $x$x$>$>$\editable{}$ |
What is the domain of the function?
$x\ge12$x≥12
$x\ge-3$x≥−3
$x\le-3$x≤−3
all real $x$x
$x\ge12$x≥12
$x\ge-3$x≥−3
$x\le-3$x≤−3
all real $x$x
What is the range of the function? Give your answer as an inequality.
Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs
Apply graphical methods in solving problems