At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Sure, let's analyze the ranges and functions carefully step-by-step. We need to determine the values in the range of [tex]\( f(x) = \sqrt{x+1} - 1 \)[/tex] that do not fall within the given value ranges of [tex]\( g(x) \)[/tex].
1. Determine the domain and range of [tex]\( f(x) \)[/tex]
- The function [tex]\( f(x) = \sqrt{x+1} - 1 \)[/tex] is defined for values of [tex]\( x \)[/tex] such that the argument of the square root is non-negative.
- Therefore, [tex]\( x + 1 \geq 0 \)[/tex] which implies [tex]\( x \geq -1 \)[/tex].
- To find the range of [tex]\( f(x) \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = \sqrt{-1 + 1} - 1 = 0 - 1 = -1 \)[/tex]
- As [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(\infty\)[/tex], [tex]\( \sqrt{x+1} \)[/tex] will also increase from 0 to [tex]\(\infty\)[/tex], and therefore [tex]\( f(x) \)[/tex] will increase from [tex]\(-1\)[/tex] to [tex]\(\infty\)[/tex].
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([-1, \infty) \)[/tex].
2. Compare the range of [tex]\( f(x) \)[/tex] with the given ranges and find values that do not intersect
Let's analyze the given ranges one by one:
- [tex]\( [-1, 1) \)[/tex]: This range starts from [tex]\(-1\)[/tex] inclusive and extends to [tex]\(1\)[/tex] but not including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] is [tex]\([-1, 1)\)[/tex]. So, this range is a valid subset of [tex]\([-1, \infty)\)[/tex].
- [tex]\( (-1, 1) \)[/tex]: This range starts just above [tex]\(-1\)[/tex] and extends to but does not include [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] is [tex]\((-1, 1)\)[/tex]. So, this range is also a valid subset of [tex]\([-1, \infty)\)[/tex].
- [tex]\( (-\infty, 1] \)[/tex]: This range extends from negative infinity up to and including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] would be [tex]\([-1, 1]\)[/tex], so this also overlaps with some part of the range of [tex]\( f(x) \)[/tex].
- [tex]\( (-\infty, 1) \)[/tex]: This range extends from negative infinity up to but not including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] would be [tex]\([-1, 1)\)[/tex], which also overlaps with some part of the range of [tex]\( f(x) \)[/tex].
Thus, all given ranges overlap with the range of [tex]\( f(x) \)[/tex]. Hence, there are no values in the range of [tex]\( f(x) = \sqrt{x+1} - 1 \)[/tex] that are not in the graph of [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ -1 \][/tex]
1. Determine the domain and range of [tex]\( f(x) \)[/tex]
- The function [tex]\( f(x) = \sqrt{x+1} - 1 \)[/tex] is defined for values of [tex]\( x \)[/tex] such that the argument of the square root is non-negative.
- Therefore, [tex]\( x + 1 \geq 0 \)[/tex] which implies [tex]\( x \geq -1 \)[/tex].
- To find the range of [tex]\( f(x) \)[/tex]:
- When [tex]\( x = -1 \)[/tex], [tex]\( f(x) = \sqrt{-1 + 1} - 1 = 0 - 1 = -1 \)[/tex]
- As [tex]\( x \)[/tex] increases from [tex]\(-1\)[/tex] to [tex]\(\infty\)[/tex], [tex]\( \sqrt{x+1} \)[/tex] will also increase from 0 to [tex]\(\infty\)[/tex], and therefore [tex]\( f(x) \)[/tex] will increase from [tex]\(-1\)[/tex] to [tex]\(\infty\)[/tex].
- Therefore, the range of [tex]\( f(x) \)[/tex] is [tex]\([-1, \infty) \)[/tex].
2. Compare the range of [tex]\( f(x) \)[/tex] with the given ranges and find values that do not intersect
Let's analyze the given ranges one by one:
- [tex]\( [-1, 1) \)[/tex]: This range starts from [tex]\(-1\)[/tex] inclusive and extends to [tex]\(1\)[/tex] but not including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] is [tex]\([-1, 1)\)[/tex]. So, this range is a valid subset of [tex]\([-1, \infty)\)[/tex].
- [tex]\( (-1, 1) \)[/tex]: This range starts just above [tex]\(-1\)[/tex] and extends to but does not include [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] is [tex]\((-1, 1)\)[/tex]. So, this range is also a valid subset of [tex]\([-1, \infty)\)[/tex].
- [tex]\( (-\infty, 1] \)[/tex]: This range extends from negative infinity up to and including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] would be [tex]\([-1, 1]\)[/tex], so this also overlaps with some part of the range of [tex]\( f(x) \)[/tex].
- [tex]\( (-\infty, 1) \)[/tex]: This range extends from negative infinity up to but not including [tex]\(1\)[/tex]. The intersection with [tex]\([-1, \infty)\)[/tex] would be [tex]\([-1, 1)\)[/tex], which also overlaps with some part of the range of [tex]\( f(x) \)[/tex].
Thus, all given ranges overlap with the range of [tex]\( f(x) \)[/tex]. Hence, there are no values in the range of [tex]\( f(x) = \sqrt{x+1} - 1 \)[/tex] that are not in the graph of [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ -1 \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
