[tex]y=\sqrt{3 x^2+4 x-1}[/tex]
Answer (2)
The domain of the function y = 3 x 2 + 4 x − 1 is ( − ∞ , 3 − 2 − 7 ] ∪ [ 3 − 2 + 7 , ∞ ) and the range is [ 0 , ∞ ) .
;
Find the domain by solving the inequality 3 x 2 + 4 x − 1 ≥ 0 .
Calculate the roots of the quadratic equation 3 x 2 + 4 x − 1 = 0 using the quadratic formula: x = 3 − 2 ± 7 .
Determine the intervals where 3 x 2 + 4 x − 1 ≥ 0 , which gives the domain: ( − ∞ , 3 − 2 − 7 ] ∪ [ 3 − 2 + 7 , ∞ ) .
The range of the function is [ 0 , ∞ ) .
Explanation
Problem Analysis We are given the function y = 3 x 2 + 4 x − 1 and asked to analyze it. This involves finding the domain, range, and general behavior of the function.
Finding the Domain First, let's find the domain of the function. Since we have a square root, the expression inside the square root must be non-negative. Therefore, we need to solve the inequality 3 x 2 + 4 x − 1 ≥ 0 .
Finding the Roots To solve this inequality, we first find the roots of the quadratic equation 3 x 2 + 4 x − 1 = 0 . We can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c , where a = 3 , b = 4 , and c = − 1 .
Calculating the Roots Plugging in the values, we get: x = 2 ( 3 ) − 4 ± 4 2 − 4 ( 3 ) ( − 1 ) = 6 − 4 ± 16 + 12 = 6 − 4 ± 28 = 6 − 4 ± 2 7 = 3 − 2 ± 7 So the roots are x 1 = 3 − 2 − 7 ≈ − 1.5486 and x 2 = 3 − 2 + 7 ≈ 0.2153 .
Determining the Domain Now we analyze the inequality 3 x 2 + 4 x − 1 ≥ 0 . Since the coefficient of x 2 is positive, the parabola opens upwards. Thus, 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 domain of the function is x ≤ 3 − 2 − 7 or x ≥ 3 − 2 + 7 . In interval notation, the domain is ( − ∞ , 3 − 2 − 7 ] ∪ [ 3 − 2 + 7 , ∞ ) .
Finding the Range To find the range, we note that since the square root function always returns non-negative values, y ≥ 0 . The vertex of the parabola 3 x 2 + 4 x − 1 occurs at x = − 2 a b = − 2 ( 3 ) 4 = − 3 2 . The value of the quadratic at this point is 3 ( − 3 2 ) 2 + 4 ( − 3 2 ) − 1 = 3 ( 9 4 ) − 3 8 − 1 = 3 4 − 3 8 − 1 = − 3 4 − 1 = − 3 7 . Since this value is negative, the minimum value of 3 x 2 + 4 x − 1 in the domain is 0, which occurs at the roots. Therefore, the range of the function is [ 0 , ∞ ) .
Final Answer In summary, the domain of the function is ( − ∞ , 3 − 2 − 7 ] ∪ [ 3 − 2 + 7 , ∞ ) , and the range is [ 0 , ∞ ) .
Examples
Understanding the domain and range of functions like this is crucial in many real-world applications. For example, if this function represented the height of a projectile over time (where x is time), knowing the domain tells us when the projectile is in the air, and the range tells us the possible heights it can reach. Similarly, in engineering, this could model the stress on a material, where the domain represents the allowable range of a parameter, and the range represents the possible stress levels.
