Question

1. Graph the equation y = -2/x – 31 + 1. State the domain and range as well as the intervals over which the function is increasing and decreasing. Domain: 17 6 Range: 5 4 3 2 Increasing interval: 19 - - 5-4 -3-2 "1 4 5 6 8 9 N Decreasing interval: 2 85 US 2

Answer 1

The graph of the absolute value function is

Then, using the rules of transformation of functions you can graph the given function. Some of these rules are:

*If you have -f(x) the function is reflected over the x-axis.

*If you have f(x-h) the function moves h units to the right.

*If you have f(x) + k the function moves k units up

*If you have a*f(x), where a > 1, the function is stretched vertically by a factor of a.

So, in this case

`\begin{gathered} y=-2|x-3|+1 \\ h=3 \\ k=1 \\ a=2 \end{gathered}`

And the graph will be

Now, by definition, the domain of a function f(x) is the set of all values for which the function is defined, so since the absolute value function is defined for all reals, then the domain of this function will be

`D_f=(-\infty,\infty)`

On the other hand, the range of the function is the set of all values that y or f(x) takes. So, as you can see in the graph, the maximum value that the function takes is 1, then the range of this function will be

`R_f=(-\infty,1\rbrack`

On the other hand, As you can see in the graph, the function has an increasing interval for the x whose values ​​are in

`(-\infty,1\rbrack`

And the function has a decreasing interval for the x whose values ​​are in

`\lbrack1,\infty)`

Some rules of transformation of functions:

*If you have -f(x) the function is reflected over the x-axis.

*If you have f(x-h) the function moves h units to the right.

*If you have f(x) + k the function moves k units up

*If you have a*f(x), where a > 1, the function is stretched vertically by a factor of a.