Join the IDNLearn.com community and start getting the answers you need today. Get prompt and accurate answers to your questions from our experts who are always ready to help.
Sagot :
To analyze the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] based on the given values, let's break down the steps:
1. Given Data:
- The table provides the values for [tex]\( g(x) \)[/tex] at specific points:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -5 & 4 \\ -4 & 1 \\ -3 & 0 \\ \hline -2 & 1 \\ \hline \end{array} \][/tex]
2. Assumption about [tex]\( f(x) \)[/tex]:
- Let's assume [tex]\( f(x) \)[/tex] is given by the function [tex]\( f(x) = x + 1 \)[/tex].
3. Calculating [tex]\( f(x) \)[/tex]:
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -5 + 1 = -4 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = -4 + 1 = -3 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -3 + 1 = -2 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -2 + 1 = -1 \][/tex]
Thus, the values for [tex]\( f(x) \)[/tex] at these points are:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -5 & -4 \\ -4 & -3 \\ -3 & -2 \\ \hline -2 & -1 \\ \hline \end{array} \][/tex]
4. Comparing [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- At [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -4, \quad g(-5) = 4 \][/tex]
- At [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = -3, \quad g(-4) = 1 \][/tex]
- At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -2, \quad g(-3) = 0 \][/tex]
- At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -1, \quad g(-2) = 1 \][/tex]
5. Checking Intersections:
- The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect where [tex]\( f(x) = g(x) \)[/tex].
- From the above comparisons, we note that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] do not intersect at any of these points.
- Therefore, there are no points where they intersect.
6. Checking [tex]\( f(x) \)[/tex] Greater than [tex]\( g(x) \)[/tex]:
- To determine if [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for all given [tex]\( x \)[/tex]:
- At [tex]\( x = -5 \)[/tex]: [tex]\( f(-5) = -4 \)[/tex] and [tex]\( g(-5) = 4 \)[/tex]. Here, [tex]\( f(-5) < g(-5) \)[/tex].
- At [tex]\( x = -4 \)[/tex]: [tex]\( f(-4) = -3 \)[/tex] and [tex]\( g(-4) = 1 \)[/tex]. Here, [tex]\( f(-4) < g(-4) \)[/tex].
- At [tex]\( x = -3 \)[/tex]: [tex]\( f(-3) = -2 \)[/tex] and [tex]\( g(-3) = 0 \)[/tex]. Here, [tex]\( f(-3) < g(-3) \)[/tex].
- At [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -1 \)[/tex] and [tex]\( g(-2) = 1 \)[/tex]. Here, [tex]\( f(-2) < g(-2) \)[/tex].
- Hence, [tex]\( f(x) \)[/tex] is not greater than [tex]\( g(x) \)[/tex] for any given [tex]\( x \)[/tex].
Conclusion:
- The statement "f(x) and g(x) intersect at exactly one point" is false.
- The statement "f(x) is greater than g(x) for all values of x" is also false.
Thus, neither statement about the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is true.
1. Given Data:
- The table provides the values for [tex]\( g(x) \)[/tex] at specific points:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -5 & 4 \\ -4 & 1 \\ -3 & 0 \\ \hline -2 & 1 \\ \hline \end{array} \][/tex]
2. Assumption about [tex]\( f(x) \)[/tex]:
- Let's assume [tex]\( f(x) \)[/tex] is given by the function [tex]\( f(x) = x + 1 \)[/tex].
3. Calculating [tex]\( f(x) \)[/tex]:
- For [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -5 + 1 = -4 \][/tex]
- For [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = -4 + 1 = -3 \][/tex]
- For [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -3 + 1 = -2 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -2 + 1 = -1 \][/tex]
Thus, the values for [tex]\( f(x) \)[/tex] at these points are:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -5 & -4 \\ -4 & -3 \\ -3 & -2 \\ \hline -2 & -1 \\ \hline \end{array} \][/tex]
4. Comparing [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- At [tex]\( x = -5 \)[/tex]:
[tex]\[ f(-5) = -4, \quad g(-5) = 4 \][/tex]
- At [tex]\( x = -4 \)[/tex]:
[tex]\[ f(-4) = -3, \quad g(-4) = 1 \][/tex]
- At [tex]\( x = -3 \)[/tex]:
[tex]\[ f(-3) = -2, \quad g(-3) = 0 \][/tex]
- At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -1, \quad g(-2) = 1 \][/tex]
5. Checking Intersections:
- The functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] intersect where [tex]\( f(x) = g(x) \)[/tex].
- From the above comparisons, we note that [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] do not intersect at any of these points.
- Therefore, there are no points where they intersect.
6. Checking [tex]\( f(x) \)[/tex] Greater than [tex]\( g(x) \)[/tex]:
- To determine if [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for all given [tex]\( x \)[/tex]:
- At [tex]\( x = -5 \)[/tex]: [tex]\( f(-5) = -4 \)[/tex] and [tex]\( g(-5) = 4 \)[/tex]. Here, [tex]\( f(-5) < g(-5) \)[/tex].
- At [tex]\( x = -4 \)[/tex]: [tex]\( f(-4) = -3 \)[/tex] and [tex]\( g(-4) = 1 \)[/tex]. Here, [tex]\( f(-4) < g(-4) \)[/tex].
- At [tex]\( x = -3 \)[/tex]: [tex]\( f(-3) = -2 \)[/tex] and [tex]\( g(-3) = 0 \)[/tex]. Here, [tex]\( f(-3) < g(-3) \)[/tex].
- At [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -1 \)[/tex] and [tex]\( g(-2) = 1 \)[/tex]. Here, [tex]\( f(-2) < g(-2) \)[/tex].
- Hence, [tex]\( f(x) \)[/tex] is not greater than [tex]\( g(x) \)[/tex] for any given [tex]\( x \)[/tex].
Conclusion:
- The statement "f(x) and g(x) intersect at exactly one point" is false.
- The statement "f(x) is greater than g(x) for all values of x" is also false.
Thus, neither statement about the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is true.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.
