What are the domain and range of the function y equals 2 start root 3x plus 4 end root minus 5?
A. lower x greater-than-or-equal-to negative four-thirds semi-colon lower y greater-than-or-equal-to negative 5
B. lower x greater-than-or-equal-to four-thirds semi-colon lower y greater-than-or-equal-to negative 5
C. lower x less-than-or-equal-to negative four-thirds semi-colon lower y less-than-or-equal-to negative 5
D. lower x great-than-or-equal-to negative four-thirds semi-colon lower y greater-than-or-equal-to 5
Well, well, well, let's dive into the world of domain and range with a splash of humor!
First things first, let's start with the domain. The domain is like a VIP list that specifies who gets access to the function. In this case, we have a square root involved, and remember, square roots aren't friendly with negative numbers. To ensure a non-imaginary result, we want the stuff inside the square root, which is 3x + 4, to be greater than or equal to zero.
Solving 3x + 4 >= 0 gives us x >= -4/3. So, our domain is all x that are greater than or equal to -4/3.
Now, let's move on to the range. The range is like the collection of presents that the function gives us as output. In this case, we don't have any restrictions on the output, so the range is all y.
Therefore, the correct answer is A. So, enjoy the ride with x greater than or equal to -4/3 and y greater than or equal to -5. Have fun!
To find the domain and range of the function y = 2√(3x + 4) - 5, we need to determine the valid values for x and the corresponding values for y.
Domain:
Since we have a square root function, the quantity inside the square root, 3x + 4, must be greater than or equal to zero.
3x + 4 ≥ 0
To solve for x, we subtract 4 from both sides of the inequality:
3x ≥ -4
Then we divide by 3 to isolate x:
x ≥ -4/3
So, the domain of the function is x ≥ -4/3.
Range:
To determine the range, we need to consider the possible values of y.
The square root function ensures that y is always greater than or equal to zero, so the function's minimum value is zero.
To find the maximum value of y, we need to consider the maximum value of the expression inside the square root, 3x + 4.
Since x can be any real number greater than or equal to -4/3, the minimum value of 3x + 4 occurs when x = -4/3.
Substituting x = -4/3 into the expression, we get:
3(-4/3) + 4 = -4 + 4 = 0
Therefore, the maximum value of 3x + 4 is 0.
Multiplying by 2 and subtracting 5:
2(0) - 5 = -5
So, the range of the function is y ≥ -5.
Putting it all together, the correct answer is:
A. x ≥ -4/3; y ≥ -5
To find the domain and range of the given function, we need to consider the restrictions on both x and y values.
Starting with the domain, we know that the expression inside the square root must be greater than or equal to zero because the square root of a negative number is undefined. So we set up the inequality:
3x + 4 ≥ 0
Now, solving for x:
3x ≥ -4
x ≥ -4/3
Thus, the domain of the function is x ≥ -4/3.
Moving on to the range, we need to consider the possible values of y. Since the expression inside the square root can be any real number, we know that the range will be all real numbers. However, there is an additional restriction because of the "-5" term at the end.
The "-5" term is a vertical translation that shifts the graph of the function down 5 units. This means that the y-values of the function will always be 5 units less than the corresponding y-values of the square root function.
Since the square root function has a range of y ≥ 0, the range of the given function will be y ≥ -5.
Therefore, the correct answer is A. lower x ≥ -4/3; lower y ≥ -5.
geez - ever hear of actually typing in algebra notation?
y = 2√(3x+4) - 5
You need a domain where 3x+4 ≥ 0
and since y is increasing, the range is
[-5, ∞)