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Sagot :
To analyze the function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex], we need to understand its transformations and how they affect the domain and range of the function. Let's break it down step by step:
1. Identify the Basic Function:
The basic function here is the square root function, [tex]\( \sqrt{x} \)[/tex].
2. Horizontal Translation:
The term [tex]\( x - 7 \)[/tex] inside the square root indicates a horizontal translation. Specifically, it shifts the graph 7 units to the right.
3. Vertical Stretch and Reflection:
The coefficient -2 in front of the square root indicates two transformations:
- The negative sign ( - ) reflects the graph across the x-axis.
- The 2 indicates a vertical stretch by a factor of 2.
4. Vertical Translation:
The term [tex]\( +1 \)[/tex] at the end indicates a vertical translation. This shifts the entire graph up by 1 unit.
### Domain Calculation
To find the domain, we need to ensure that the expression inside the square root is non-negative:
[tex]\[ x - 7 \geq 0 \][/tex]
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of the function is all [tex]\( x \)[/tex] values from 7 to infinity:
[tex]\[ \text{Domain: } [7, \infty) \][/tex]
### Range Calculation
To find the range, we analyze the output values of the function:
- The square root function [tex]\( \sqrt{x-7} \)[/tex] yields non-negative outputs (i.e., [tex]\( \sqrt{x-7} \geq 0 \)[/tex]).
- When multiplied by -2, [tex]\( -2 \sqrt{x-7} \)[/tex] produces non-positive outputs (i.e., [tex]\( -2 \sqrt{x-7} \leq 0 \)[/tex]).
- The maximum value of [tex]\( -2 \sqrt{x-7} \)[/tex] is 0 (when [tex]\( x = 7 \)[/tex]).
- The minimum value of [tex]\( -2 \sqrt{x-7} \)[/tex] approaches negative infinity as [tex]\( x \)[/tex] increases.
- Adding 1 to [tex]\( -2 \sqrt{x-7} \)[/tex] (i.e., [tex]\( -2 \sqrt{x-7} + 1 \)[/tex]) will raise all values by 1.
Thus, the maximum value of the function [tex]\( f(x) \)[/tex] is 1 (when [tex]\( x = 7 \)[/tex]) and it decreases without bound as [tex]\( x \)[/tex] increases.
Hence, the range of the function is:
[tex]\[ \text{Range: } (-\infty, 1] \][/tex]
### Conclusion
The function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] is a horizontally shifted, vertically stretched and reflected, and vertically translated square root function.
Its domain is:
[tex]\[ [7, \infty) \][/tex]
And its range is:
[tex]\[ (-\infty, 1] \][/tex]
So, the statement that best describes [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] is that it has a domain of [tex]\( [7, \infty) \)[/tex] and a range of [tex]\( (-\infty, 1] \)[/tex].
1. Identify the Basic Function:
The basic function here is the square root function, [tex]\( \sqrt{x} \)[/tex].
2. Horizontal Translation:
The term [tex]\( x - 7 \)[/tex] inside the square root indicates a horizontal translation. Specifically, it shifts the graph 7 units to the right.
3. Vertical Stretch and Reflection:
The coefficient -2 in front of the square root indicates two transformations:
- The negative sign ( - ) reflects the graph across the x-axis.
- The 2 indicates a vertical stretch by a factor of 2.
4. Vertical Translation:
The term [tex]\( +1 \)[/tex] at the end indicates a vertical translation. This shifts the entire graph up by 1 unit.
### Domain Calculation
To find the domain, we need to ensure that the expression inside the square root is non-negative:
[tex]\[ x - 7 \geq 0 \][/tex]
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of the function is all [tex]\( x \)[/tex] values from 7 to infinity:
[tex]\[ \text{Domain: } [7, \infty) \][/tex]
### Range Calculation
To find the range, we analyze the output values of the function:
- The square root function [tex]\( \sqrt{x-7} \)[/tex] yields non-negative outputs (i.e., [tex]\( \sqrt{x-7} \geq 0 \)[/tex]).
- When multiplied by -2, [tex]\( -2 \sqrt{x-7} \)[/tex] produces non-positive outputs (i.e., [tex]\( -2 \sqrt{x-7} \leq 0 \)[/tex]).
- The maximum value of [tex]\( -2 \sqrt{x-7} \)[/tex] is 0 (when [tex]\( x = 7 \)[/tex]).
- The minimum value of [tex]\( -2 \sqrt{x-7} \)[/tex] approaches negative infinity as [tex]\( x \)[/tex] increases.
- Adding 1 to [tex]\( -2 \sqrt{x-7} \)[/tex] (i.e., [tex]\( -2 \sqrt{x-7} + 1 \)[/tex]) will raise all values by 1.
Thus, the maximum value of the function [tex]\( f(x) \)[/tex] is 1 (when [tex]\( x = 7 \)[/tex]) and it decreases without bound as [tex]\( x \)[/tex] increases.
Hence, the range of the function is:
[tex]\[ \text{Range: } (-\infty, 1] \][/tex]
### Conclusion
The function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] is a horizontally shifted, vertically stretched and reflected, and vertically translated square root function.
Its domain is:
[tex]\[ [7, \infty) \][/tex]
And its range is:
[tex]\[ (-\infty, 1] \][/tex]
So, the statement that best describes [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] is that it has a domain of [tex]\( [7, \infty) \)[/tex] and a range of [tex]\( (-\infty, 1] \)[/tex].
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