Determine the domain and range of the function

Question

Determine the domain and range of the function y=√(x²+7x+10)

Answer 1

Let's first look at the domain of the function.  When we are looking for the domain of a function, we are looking for what values of x can we plug into the function and get a real output value (or y value).  This is an interesting function because it is a composite function.  The outside function is the square root function while the inside function is the quadratic x²+7x+10.  The square root piece is the operation that causes our domain issues.  We can't take the square root of a negative number.  This means that if we first focus on what values of x give us a negative output for x²+7x+10, we will be able to eliminate them from the domain of the overall function.  

 

So let's look at x²+7x+10.  We want to know where this parabola takes on value above and below the x axis.  Wherever it takes on value above the x axis, it will take on value for our composite function (again, the composite function is the original function you posed in your original question).  So where does x²+7x+10 = 0?

 

x²+7x+10 = 0

(factor)

(x+2)(x+5) = 0

(zero product property)

(x+2) = 0  and  (x+5) = 0

(Solve for x in each mini-equation)

x = -2 and x = -5

 

So this means that the parabola produced by the equation x²+7x+10 crosses the x-axis at x = -5 and -2.  Since this parabola opens up, we know that the parabola will be above the x-axis for all values smaller than x = -5.  The parabola will also be above the x-axis for all values of x that are greater than -2.  

 

So if you plug in any x value smaller than -5, bigger than -2 or equal to -5 or -2 into x²+7x+10 you get an output that is either equal to 0 or greater than 0.  If you plug in a value for x that is between -5 and -2 you get a negative output.  Therefore, we can only use the x values that produce outputs equal to 0 or greater.  So the domain of the original function is any x smaller than or equal to -5 or any x bigger than or equal to -2.

 

Now let's talk about the range.  The range of a function are the possible output values for that function.  We commonly refer to the output values of a function as the y values of a function.  So, the values that we can plug into the original function produce values that are greater than or equal to 0.  So therefore, our range for this function is any y greater than or equal to zero.

 

Please let me know if this helps or if I can help in any other way.

 

Thank you,

Tyler