What is the domain of the function y=3 sq root 6x+42 ?
x>=0
x<=7
x>=-6
x>=-7
The domain of the function is x>=-7.
This is because the expression inside the square root, 6x+42, must be non-negative for real values of y to exist.
Setting 6x+42>=0 and solving for x, we get x>=-7. Therefore, the domain of the function is x>=-7.
2. What are the domain and range of the function (Image: y equals 2 start root 3x plus 4 end root minus 5)? (1 point)
(Image: lower x greater-than-or-equal-to negative four-thirds semi-colon lower y greater-than-or-equal-to negative 5)
(Image: lower x greater-than-or-equal-to four-thirds semi-colon lower y greater-than-or-equal-to negative 5)
(Image: lower x less-than-or-equal-to negative four-thirds semi-colon lower y less-than-or-equal-to negative 5)
(Image: lower x great-than-or-equal-to negative four-thirds semi-colon lower y greater-than-or-equal-to 5)
The domain of the function is x >= -4/3.
To find the domain, we need to consider the expression inside the square root, 3x + 4. For real values of y to exist, 3x + 4 must be non-negative (or equal to zero) so that we do not take the square root of a negative number. Solving 3x + 4 >= 0, we get x >= -4/3.
The range of the function is y >= -5.
To see this, note that the square root term is always non-negative since we only deal with the positive square root. Therefore, the minimum value of y occurs when the expression inside the square root is zero, which occurs when 3x + 4 = 0, or x = -4/3. Evaluating the function at this point, we get y = -5. Since y is always greater than or equal to -5 (but can be greater when x is larger), the range of the function is y >= -5.
3. Which of the following is a graph of the equation (Image: y equals StartRoot x plus 5 EndRoot minus 2)?
(Image: The graph shows the equation y equals StartRoot x EndRoot translated 5 units to the left and 2 units down.)
(Image: The graph shows the equation y equals StartRoot x EndRoot translated 5 units to the left and 2 units up.)
(Image: The graph shows the equation y equals StartRoot x EndRoot translated 5 units to the right and 2 units up.)
(Image: The graph shows the equation y equals StartRoot x EndRoot translated 5 units to the right and 2 units down.)
The graph that shows the equation y equals StartRoot x plus 5 EndRoot minus 2 is:
(Image: The graph shows the equation y equals StartRoot x EndRoot translated 5 units to the left and 2 units down.)
To obtain this graph, we start with the graph of y = sqrt(x), which is a curve that starts at the origin and moves upwards and to the right. To obtain y = sqrt(x + 5) - 2, we first shift the graph 5 units to the left by replacing x with (x + 5), as follows: y = sqrt(x + 5). This moves the graph to the left by 5 units. Then, we shift the graph down 2 units by subtracting 2 from the equation: y = sqrt(x + 5) - 2. This moves the graph down by 2 units. The resulting curve has its vertex at (-5, -2) and moves upwards and to the right.
To determine the domain of a function, we need to identify the set of possible values for the input variable (x) for which the function is defined. In this case, we are given some specific conditions for x.
The function is y = 3√(6x + 42). We have the following conditions mentioned:
1. x >= 0
2. x <= 7
3. x >= -6
4. x >= -7
To find the domain, we need to consider all the given conditions and find the common range of values that satisfy all the conditions.
Condition 1 (x >= 0): This condition states that x must be greater than or equal to 0. It means that x is a nonnegative number or zero.
Condition 2 (x <= 7): This condition states that x must be less than or equal to 7. It means that x is a number that is at most 7.
Condition 3 (x >= -6): This condition states that x must be greater than or equal to -6. It means that x is a number that is at least -6.
Condition 4 (x >= -7): This condition states that x must be greater than or equal to -7. It means that x is a number that is at least -7.
To find the common range of values that satisfy all the given conditions, we need to find the intersection of these conditions.
From the given conditions, we can conclude that x must be greater than or equal to 0 (x >= 0) and less than or equal to 7 (x <= 7).
Therefore, the domain of the function y = 3√(6x + 42) is x >= 0 and x <= 7.