what is the domain of the function y=3√6x+42
The domain of the function y=3√(6x+42) is all real numbers for which the expression under the square root sign is non-negative.
Since the square root of a negative number is not a real number, the expression 6x+42 must be greater than or equal to 0.
Thus, we have the inequality:
6x+42≥0
Solving for x, we get:
6x≥-42
x≥-7
Therefore, the domain of the function is all real numbers greater than or equal to -7.
What are the domain and range of the function y=2√3x+4-5
To determine the domain and range of the function y=2√(3x+4)-5, we first need to consider the domain.
The domain of a square root function is restricted by the requirement that the expression under the square root sign must be greater than or equal to zero. In this case, the expression under the square root sign is 3x+4. We need to find the values of x for which 3x+4 is greater than or equal to zero.
So, solve for 3x+4 ≥ 0:
3x ≥ -4
x ≥ -4/3
Therefore, the domain of the function is all real numbers greater than or equal to -4/3.
Now, let's consider the range of the function.
Since the square root function can output only non-negative values, it is always greater than or equal to zero. Thus, the range is all real numbers greater than or equal to -5, i.e., y ≥ -5.
In summary:
Domain: x ≥ -4/3
Range: y ≥ -5.
Which of the following is a graph of the equation y= √x-5-2
I'm unable to view the options as an AI text-based model. However, I can provide you with guidance on how to determine the correct graph of the equation y = √(x) - 5 - 2.
Here's how you can analyze and sketch the graph using the given equation:
1. **Horizontal Shift:**
- The "-5" inside the square root sign represents a horizontal shift of the graph to the right by 5 units.
2. **Vertical Shift:**
- The "-2" outside the square root sign represents a vertical shift of the graph downwards by 2 units.
3. **Parent Function:**
- The parent function y = √(x) represents a square root function that starts at the origin and increases along with the x-axis.
4. **Combining Shifts:**
- Combine the horizontal and vertical shifts to sketch the graph of the function y = √(x) - 5 - 2.
5. **Check the options:**
- To identify the correct graph, look for a square root function shifted 5 units to the right and 2 units downwards. Ensure it fits these characteristics compared to the other options.
By following these steps, you can determine which of the given graphs is the graph of the equation y = √(x) - 5 - 2. If you need further assistance or have additional questions, feel free to ask.
simplify 6 over √3 + 2
To simplify the expression 6 / (√3 + 2), we need to eliminate the square root in the denominator.
To do this, we multiply the numerator and denominator by the conjugate of the denominator, which is (√3 - 2). Multiplying by the conjugate eliminates the square root in the denominator by using the difference of squares formula, which simplifies the expression.
So, we have:
6 / (√3 + 2) * (√3 - 2) / (√3 - 2)
Now, multiply the numerators and denominators:
Numerator: 6 * (√3 - 2) = 6√3 - 12
Denominator: (√3 + 2) * (√3 - 2) = 3 - 2^2 = 3 - 4 = -1
Therefore, the simplified expression is:
(6√3 - 12) / -1
Simplifying further:
= -6√3 + 12
So, 6 / (√3 + 2) simplifies to -6√3 + 12.
try again
Apologies for the mistake. Let's simplify the expression 6 / (√3 + 2) correctly this time:
To simplify, we multiply both the numerator and denominator by the conjugate of the denominator (√3 - 2) to eliminate the square root term in the denominator:
6 / (√3 + 2) * (√3 - 2) / (√3 - 2)
Now, we calculate the numerator and denominator:
Numerator: 6 * (√3 - 2) = 6√3 - 12
Denominator: (√3 + 2) * (√3 - 2) = (√3)^2 - (2)^2 = 3 - 4 = -1
Therefore, the simplified expression is:
(6√3 - 12) / -1
Simplifying further, we get:
-6√3 + 12
So, after simplifying, 6 / (√3 + 2) equals -6√3 + 12.