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Magazine Chapter 7: Problem 18
Graph each function. State the domain and range of each function. \(y=5-\sqrt{x+4}\)
Short Answer
Expert verified
Domain: [-4, 鈭�); Range: (-鈭�, 5]
Step by step solution
01
Identify the Function Type
The given function is \( y = 5 - \sqrt{x+4} \). This is a transformed square root function, which involves a vertical shift and reflection.
02
Determine the Domain
The square root function \( \sqrt{x+4} \) is only defined for \( x+4 \geq 0 \), which means \( x \geq -4 \). Therefore, the domain in interval notation is \([-4, \infty)\).
03
Determine the Range
Since \( y = 5 - \sqrt{x+4} \), the maximum value of \( \sqrt{x+4} \) is 0 (at \( x = -4 \)) and decreases from there as \( x \) increases. Thus, \( y \) has a maximum value of 5. The range of the function is \((-fty, 5] \).
04
Sketch the Function
Start by plotting the key point \((-4, 5)\) since when \( x = -4 \), \( y \) is 5. As \( x \) increases, plot additional points such as when \( x = 0, y = 3 \), building a curve extending downward to the right. The curve reflects over the horizontal line through \( y = 5 \).
05
Conclude with the Full Graph
Draw a smooth curve gradually decreasing from the point \((-4, 5)\) through the plotted points, ensuring the graph does not fall below 5 anywhere across its domain. This completes the graph of \( y = 5 - \sqrt{x+4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Domain of a Function
When considering a function like \( y = 5 - \sqrt{x+4} \), the domain is all the permissible values of \( x \). To determine the domain for this specific function, you must consider the expression inside the square root: \( \sqrt{x+4} \).
This part of the mathematical expression is only defined when \( x+4 \geq 0 \). Thus, we find \( x \geq -4 \). This means any value of \( x < -4 \) would result in taking the square root of a negative number, which isn't possible with real numbers. Therefore, the domain of the function is restricted to:
This tells us \( x \) can be as small as \(-4 \), going infinitely in the positive direction.
This part of the mathematical expression is only defined when \( x+4 \geq 0 \). Thus, we find \( x \geq -4 \). This means any value of \( x < -4 \) would result in taking the square root of a negative number, which isn't possible with real numbers. Therefore, the domain of the function is restricted to:
- \([-4, \infty)\) in interval notation.
This tells us \( x \) can be as small as \(-4 \), going infinitely in the positive direction.
Range of a Function
The range of a function is all the possible values that \( y \) can take. For our transformed square root function \( y = 5 - \sqrt{x+4} \), understanding how the function behaves helps us determine its range.
Initially, consider the square root component \( \sqrt{x+4} \). As we've learned earlier, this expression is defined for \( x > -4 \). The value of \( \sqrt{x+4} \) starts at \( 0 \) when \( x = -4 \) and increases as \( x \) becomes larger. However, since our whole function is \( 5 - \sqrt{x+4} \), it means the function starts at a maximum of \( 5 \) when \( x \) is \(-4\).
As \( x \) increases, \( \sqrt{x+4} \) becomes larger, making the whole \( 5 - \sqrt{x+4} \) expression smaller. This demonstrates that as \( x \) becomes large, \( y \) decreases continuously, but it never goes below any particular value since \( \sqrt{x+4} \) grows indefinitely.
Thus, the range is:
\( y \) can take any value less than or equal to 5.
Initially, consider the square root component \( \sqrt{x+4} \). As we've learned earlier, this expression is defined for \( x > -4 \). The value of \( \sqrt{x+4} \) starts at \( 0 \) when \( x = -4 \) and increases as \( x \) becomes larger. However, since our whole function is \( 5 - \sqrt{x+4} \), it means the function starts at a maximum of \( 5 \) when \( x \) is \(-4\).
As \( x \) increases, \( \sqrt{x+4} \) becomes larger, making the whole \( 5 - \sqrt{x+4} \) expression smaller. This demonstrates that as \( x \) becomes large, \( y \) decreases continuously, but it never goes below any particular value since \( \sqrt{x+4} \) grows indefinitely.
Thus, the range is:
- \(( -\infty, 5 ]\)
\( y \) can take any value less than or equal to 5.
Graphing Functions
Graphing this function, \( y = 5 - \sqrt{x+4} \), involves plotting points and connecting them to reveal the general shape of the graph. Begin by identifying important points that serve as a guide for drawing this curve.
The key starting point, also considered the vertex in this context, is \( (-4, 5) \). Here, at \( x = -4 \), the value of \( y \) is at its maximum, \( 5 \). As \( x \) increases from this point, calculate additional points to assist in sketching: for instance, at \( x=0 \), we find \( y=3 \).
Mark these on your graph, and draw a curve connecting them, starting from the vertex downward to the right. The curve should gradually decrease smoothly, reflecting vertically from a horizontal line at \( y = 5 \).
Remember:
By following these steps, you effectively plot the transformed square root function on a coordinate plane, visualizing its domain and range.
The key starting point, also considered the vertex in this context, is \( (-4, 5) \). Here, at \( x = -4 \), the value of \( y \) is at its maximum, \( 5 \). As \( x \) increases from this point, calculate additional points to assist in sketching: for instance, at \( x=0 \), we find \( y=3 \).
Mark these on your graph, and draw a curve connecting them, starting from the vertex downward to the right. The curve should gradually decrease smoothly, reflecting vertically from a horizontal line at \( y = 5 \).
Remember:
- The graph never goes above \( y = 5 \)
- The slope is always descending as \( x \) increases
By following these steps, you effectively plot the transformed square root function on a coordinate plane, visualizing its domain and range.
