1/(((5x^2)+(4x)-1)^(1/2))

Question

 I would like to know the domain for the function please. and just to clarify, the bottom is the square root of 5x squared plus 4x minus 1

Answer 1

Your use of parentheses was pretty good.  And to enforce your meaning through translation was also good.  Proof that you understand the order of operations.

 

1 / [√(5x² + 4x - 1)]

  

The domain is the set of x values in which the expression or function is defined.  Set the term under the square-root greater than zero.  Create an quadratic inequality.

 

5x² + 4x - 1 > 0

 

The reason for this is because setting this equal to zero would make the whole thing undefined, and setting this less then zero would not give us a real result.

 

Now we can solve for x.  To make it easier to solve for x, lets replace the greater than sign with an equal sign.  You will see what I mean in the succeeding steps.

 

5x² + 4x - 1 = 0

 

 

Try to factor the left side using FOIL.

 

(5x - 1)(x + 1) = 0

 

 

Setting the factors equal to zero, we get

 

x = -1           and         x = 1/5

 

These are the x values on the x-axis.  Since we want the quadratic to be greater than zero, we pick a range of x values that are above the x-axis.  Being that the quadratic represents a parabola opening upward, the domain will be in the interval

 

(-∞, -1)∪(1/5, ∞)

 

As you can see, knowing the shape of a parabola helped a great deal here.  Otherwise, you would be using test points to verify the inequality to be true.

Answer 2

John, 5x² + 4x - 1 has to be positive, because the denominator can't be 0 and we can't take the square root of a negative number.

 

I can't find a number that makes it negative, so we just have to concentrate on what makes it 0.

 

(5x - 1)(x + 1) = 0

 

5x - 1 = 0

5x = 1

x = 1/5

 

x + 1 = 0

x = -1

 

So, x ≠ 1/5 or -1

 

Hope this helps!