$y=\sqrt{3 x^2+14 x-1}$
Answer (1)
Find the roots of the quadratic equation 3 x 2 + 14 x − 1 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Determine the intervals where 3 x 2 + 14 x − 1 ≥ 0 . Since the parabola opens upwards, the quadratic expression is non-negative when x is less than or equal to the smaller root or greater than or equal to the larger root.
The roots are x 1 = 3 − 7 − 2 13 and x 2 = 3 − 7 + 2 13 .
The domain of the function is x ≤ 3 − 7 − 2 13 or x ≥ 3 − 7 + 2 13 . x ≤ 3 − 7 − 2 13 or x ≥ 3 − 7 + 2 13
Explanation
Understanding the Problem The problem asks us to apply the 'USO CHAM RULE' to the function y = 3 x 2 + 14 x − 1 . However, the problem does not define what the 'USO CHAM RULE' is. Assuming that the intention of the problem is to find the domain of the function, we need to determine the values of x for which the expression inside the square root is non-negative, i.e., 3 x 2 + 14 x − 1 ≥ 0 .
Finding the Roots To find the domain, we need to solve the quadratic inequality 3 x 2 + 14 x − 1 ≥ 0 . First, we find the roots of the quadratic equation 3 x 2 + 14 x − 1 = 0 . We can use the quadratic formula to find the roots:
Applying the Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c . In this case, a = 3 , b = 14 , and c = − 1 . Plugging these values into the formula, we get:
x = 2 ( 3 ) − 14 ± 1 4 2 − 4 ( 3 ) ( − 1 ) = 6 − 14 ± 196 + 12 = 6 − 14 ± 208 = 6 − 14 ± 4 13 = 3 − 7 ± 2 13
So the roots are x 1 = 3 − 7 − 2 13 and x 2 = 3 − 7 + 2 13 .
Determining the Intervals Now we need to determine the intervals where 3 x 2 + 14 x − 1 ≥ 0 . Since the coefficient of the x 2 term is positive ( 0"> a = 3 > 0 ), the parabola opens upwards. This means that the quadratic expression is non-negative when x is less than or equal to the smaller root or greater than or equal to the larger root.
Therefore, the solution to the inequality 3 x 2 + 14 x − 1 ≥ 0 is:
x ≤ 3 − 7 − 2 13 or x ≥ 3 − 7 + 2 13
Approximating the Roots We can approximate the roots as follows:
x 1 = 3 − 7 − 2 13 ≈ 3 − 7 − 2 ( 3.606 ) ≈ 3 − 7 − 7.212 ≈ 3 − 14.212 ≈ − 4.737
x 2 = 3 − 7 + 2 13 ≈ 3 − 7 + 2 ( 3.606 ) ≈ 3 − 7 + 7.212 ≈ 3 0.212 ≈ 0.071
So the domain of the function is approximately x ≤ − 4.737 or x ≥ 0.071 .
Final Answer Therefore, the domain of the function y = 3 x 2 + 14 x − 1 is x ≤ 3 − 7 − 2 13 or x ≥ 3 − 7 + 2 13 .
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if we are modeling the height of a projectile as a function of time, the domain would represent the valid time intervals for which the height is a real number. Similarly, in business, if we are modeling profit as a function of the number of units sold, the domain would represent the number of units that can be sold to make a non-negative profit. In this case, finding the domain of the function y = 3 x 2 + 14 x − 1 helps us determine the set of possible x-values for which the function is defined, which can be applied to various scenarios where this type of function is used to model a real-world phenomenon.
