Question

Using y=1/x as the parent function, make your own transformations (5 units right, reflect on x axis, 2 units down, horizontal compression with factor 2). Then graph and state domain and range.

Answer 1

First, we have a translation of 5 units right. That means the value of x is decreased by 5 units (x' = x - 5)

So we have:

$y=\frac{1}{x-5}$

Then, we have a reflection over the x-axis, which means the value of the function changes signal (y' = -y)

$y=-\frac{1}{x-5}$

Finally, we have a translation of 2 units down, which means the value of the function is decreased by 2 units (y' = y - 2)

$y=-\frac{1}{x-5}-2$

Adding a horizontal compression by a factor of 2 means the value of x will be multiplied by 2 (x' = 2x)

$y=-\frac{1}{2x-5}-2$

Graphing this function, we have:

The domain is all values x can assume. Since we have a fraction, its denominator can't be zero, so:

$\begin{gathered} 2x-5\ne0 \\ 2x\ne5 \\ x\ne2.5 \end{gathered}$

So the domain is:

$(-\propto,2.5)\cup(2.5,\propto)$

The range is all values y can assume. Since the fraction can't have a value of zero, we have:

$\begin{gathered} y\ne0-2 \\ y\ne-2 \end{gathered}$

So the range is:

$(-\propto,-2)\cup(-2,\propto)$