What are the domain and range of the function y= 2 the square root of (3x+4) - 5
Well, the domain is where x can take any real value that makes the expression inside the square root non-negative. So, 3x + 4 must be greater than or equal to zero. Solving this inequality, we get x ≥ -4/3.
As for the range, since we have a square root function, the value inside the square root must be greater than or equal to zero for real solutions. So, 3x +4 must be greater than or equal to zero. Solving this inequality, we get that x ≥ -4/3.
So, to sum it up, the domain and range of the function are both x ≥ -4/3. It seems like we have an exclusive membership offer for the -4/3 club, where everyone is welcome!
To find the domain and range of the function y = 2√(3x + 4) - 5, we need to determine the values that x can take and the corresponding values of y.
1. Domain:
The domain of a function refers to the set of all possible values that x can take.
In this case, we have a square root function inside the expression √(3x + 4). To ensure that the square root is defined, the expression under the root (√(3x + 4)) must be non-negative. So, we set 3x + 4 ≥ 0 and solve for x:
3x + 4 ≥ 0
3x ≥ -4
x ≥ -4/3
Therefore, the domain of the function is x ≥ -4/3, since any x value greater than or equal to -4/3 makes the expression under the square root non-negative.
2. Range:
The range of a function refers to the set of all possible values that y can take.
The function starts with a square root (√(3x + 4)), which means the output (y) will always be greater than or equal to 0. Additionally, we have the term "-5" in the function, which shifts the entire graph down by 5 units.
So, the smallest possible value of y occurs when the expression √(3x + 4) is equal to 0, which happens at x = -4/3. When x is less than -4/3, the square root (√(3x + 4)) will be an imaginary number, and therefore, the function is not defined.
Therefore, the range of the function is y ≥ -5, meaning the values of y are greater than or equal to -5.
In summary:
- Domain: x ≥ -4/3
- Range: y ≥ -5
To determine the domain and range of a function, we need to consider the possible values that x and y can take.
Starting with the domain, we need to identify any restrictions on the values of x that would make the function undefined. In this case, since we have a square root function, the expression inside the square root (3x + 4) must be non-negative. Therefore, we set 3x + 4 ≥ 0 and solve for x:
3x + 4 ≥ 0
3x ≥ -4
x ≥ -4/3
So, the domain of the function is all real numbers greater than or equal to -4/3.
Moving on to the range, we need to determine the set of possible values that y can take. Since we have a square root function in the form of √(3x + 4), the expression inside the square root must be non-negative. Moreover, the function is shifted downward by 5 units due to the "-5". Hence, we need to find the minimum value that the square root can take and then subtract 5.
To find the minimum value, we set 3x + 4 equal to zero:
3x + 4 = 0
3x = -4
x = -4/3
Substituting this value of x back into the function, we find:
y = 2√(-4/3 + 4) - 5
Simplifying further:
y = 2√(8/3) - 5
y = 2(√(8)/√(3)) - 5
y = 2(2√(2)/√(3)) - 5
y = (4√2)/√3 - 5
So, the minimum value that y can take is (4√2)/√3 - 5.
To summarize, the domain of the function is x ≥ -4/3, and the range of the function is y ≥ (4√2)/√3 - 5.
x better be >/= -4/3
then y is always </= -5