# What is the domain and range of y = (x+1)/(x^2-7x+10)?

Dec 11, 2017

See below

#### Explanation:

Firstly, the domain of a function is any value of $x$ that can possibly go inside without causing any errors such as a division by zero, or a square root of a negative number.

Therefore, in this case, the domain is where the denominator is equal to $0$.
This is ${x}^{2} - 7 x + 10 = 0$
If we factorise this, we get
$\left(x - 2\right) \left(x - 5\right) = 0$
$x = 2 , \mathmr{and} x = 5$

So therefore, the domain is all values of $x$ where $x \ne 2$ and $x \ne 5$. This would be $\left\{x \in \mathbb{R} | x \ne 2 , x \ne 5\right\}$

To find the range of a rational function, you can look at its graph. To sketch a graph, you can look for its vertical/oblique/horizontal asymptotes and use a table of values.
This is the graph graph{(x+1)/(x^2-7x+10) [-2.735, 8.365, -2.862, 2.688]}

Can you see what the range is? Remember, the range of a function is how much you can get out of a function; The lowest possible $y$ value to the highest possible $y$ value.