Find the domain of the function. (Write your answer using interval notation.)
sqrt(x)/
(4x^2 + 3x − 1)
If you set the bottom equal to zero, I got that x can't be -1 or 1/4. But I'm not sure if that's right, or how to write the answer in interval notation.
Y = 4x^2 + 3x - 1. Factor it:
A*C = 4*(-1) = -4.
4x^2 + (4x-x) - 1.
4x^2+4x - (x+1).
4x(x+1) - (x+1).
Y = (x+1)(4x-1).
All real values of x will give a real
output. Therefore, the domain is all real values of x.
Interval notation:
-infinity < X < +infinity.
So x must be greater than (-)infinity but less than +infinity.
To find the domain of the function, we need to determine the values of x for which the function is defined.
You correctly identified that the function is undefined when the denominator, (4x^2 + 3x - 1), is equal to zero. Setting the denominator equal to zero, we have:
4x^2 + 3x - 1 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Since factoring might not be immediately apparent, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given equation, a = 4, b = 3, and c = -1. Substituting these values into the quadratic formula:
x = (-(3) ± √((3)^2 - 4(4)(-1))) / (2(4))
Simplifying further:
x = (-3 ± √(9 + 16)) / 8
x = (-3 ± √25) / 8
x = (-3 ± 5) / 8
This gives us two solutions:
x = ( -3 + 5) / 8 = 2 / 8 = 1/4
x = ( -3 - 5) / 8 = -8 / 8 = -1
Therefore, the domain of the function is all real numbers except x = 1/4 and x = -1. To write this in interval notation, we can use parentheses for excluded values and square brackets for included values:
Domain: (-∞, -1) ∪ (-1, 1/4) ∪ (1/4, ∞)
To find the domain of the function, we need to consider two conditions:
1. The denominator cannot be equal to zero since division by zero is undefined.
2. The square root function requires a non-negative value under the radical.
First, let's address the denominator. We can solve the equation 4x^2 + 3x - 1 = 0 by factoring or using the quadratic formula:
4x^2 + 3x - 1 = 0
Factoring, we have:
(4x - 1)(x + 1) = 0
Setting each factor equal to zero:
4x - 1 = 0 or x + 1 = 0
Solving these equations gives:
4x = 1 or x = -1
x = 1/4 or x = -1
So, as you correctly mentioned, x cannot be equal to 1/4 or -1.
Next, let's consider the square root function. We need to ensure that the expression inside the square root is non-negative. Since the numerator is a square root, we need to ensure that x ≥ 0.
Combining both conditions, we find that the domain of the function is x ∈ (-∞, -1) U (-1, 0] U (0, 1/4) U (1/4, ∞).
In interval notation, the domain is (-∞, -1) ∪ (-1, 0] ∪ (0, 1/4) ∪ (1/4, ∞).