Find the domain and range of the function g(x) = RADICAL 10+ 2X- X^2
To find the domain and range of the function g(x) = √(10 + 2x - x^2), we need to consider the restrictions on the input (x-values) and the resulting output (y-values).
1. Domain:
The domain of a function represents all the possible x-values for which the function is defined. In this case, since we have a square root (√) function, the argument inside the square root must be non-negative (greater than or equal to zero).
Setting the argument to be greater than or equal to zero, we can solve the inequality:
10 + 2x - x^2 ≥ 0
Rearranging the inequality, we have:
x^2 - 2x - 10 ≤ 0
Now, to find the domain, we can solve this quadratic inequality. We can factorize or use the quadratic formula to find the solutions and determine the intervals for which the inequality is satisfied.
The solutions of x^2 - 2x - 10 = 0 can be found using the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4·1·(-10))) / (2·1)
= (2 ± √(4 + 40)) / 2
= (2 ± √44) / 2
= 1 ± √11
From the solutions, we have x = 1 + √11 and x = 1 - √11.
Now, consider the quadratic inequality x^2 - 2x - 10 ≤ 0. We can determine the sign of the inequality in each interval by testing a point within that interval. We can choose the points x = 0, x = 1, and x = 2, for simplicity.
If we substitute x = 0, the inequality becomes:
(0)^2 - 2(0) - 10 ≤ 0
-10 ≤ 0 (True)
If we substitute x = 1, the inequality becomes:
(1)^2 - 2(1) - 10 ≤ 0
-12 ≤ 0 (True)
If we substitute x = 2, the inequality becomes:
(2)^2 - 2(2) - 10 ≤ 0
-10 ≤ 0 (True)
From these test results, we see that the inequality is satisfied for x-values between 1 - √11 and 1 + √11. Therefore, the domain of the function g(x) is:
Domain: [1 - √11, 1 + √11]
2. Range:
The range of a function represents all the possible y-values or outputs that the function can produce.
Since we have a square root (√) function, the range is determined by the minimum value of the function. To find the minimum value, we can observe that √(10 + 2x - x^2) is the square root of an expression. The square root function is always non-negative (greater than or equal to zero), so it does not produce negative y-values.
To find the minimum value, we can determine the vertex of the parabola formed by the expression inside the square root, 10 + 2x - x^2. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a = -1, b = 2, and c = 10.
x = -2 / (2·(-1)) = 1
Substituting x = 1 back into the expression, we find the minimum value:
√(10 + 2(1) - (1)^2) = √(12) = 2√3
Therefore, the range of the function g(x) is:
Range: [0, 2√3]
In summary:
Domain: [1 - √11, 1 + √11]
Range: [0, 2√3]