Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's analyze the given functions and the statements one by one.
### Function [tex]\( f(x) \)[/tex]
Function [tex]\( f \)[/tex] is defined by the values given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & -12 & -4 & 0 & 2 & 3 \\ \hline \end{array} \][/tex]
### Function [tex]\( g(x) \)[/tex]
Function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -12 \left( \frac{1}{3} \right)^x \][/tex]
### Analyzing Statements
1. The functions have the same [tex]\( y \)[/tex]-intercept.
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = -12 \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = -12 \left( \frac{1}{3} \right)^0 = -12 \][/tex]
Both functions have [tex]\( y \)[/tex]-intercept at [tex]\( (0, -12) \)[/tex]. Therefore, this statement is true.
2. The functions have the same [tex]\( x \)[/tex]-intercept.
The [tex]\( x \)[/tex]-intercept occurs when the function value is 0.
For [tex]\( f(x) \)[/tex]:
From the table, [tex]\( f(2) = 0 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercept for [tex]\( f(x) \)[/tex] is [tex]\( x = 2 \)[/tex].
For [tex]\( g(x) \)[/tex]:
[tex]\[ -12 \left( \frac{1}{3} \right)^x = 0 \][/tex]
This equation does not have any real solution because [tex]\( -12 \left( \frac{1}{3} \right)^x \)[/tex] never equals 0 for any real [tex]\( x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] has no [tex]\( x \)[/tex]-intercept.
Hence, this statement is false.
3. Both functions approach the same value as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex].
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] for [tex]\( f(x) \)[/tex], the function values seem to increase slowly based on the table. For [tex]\( g(x) \)[/tex]:
[tex]\[ \lim_{x \to \infty} g(x) = \lim_{x \to \infty} -12 \left( \frac{1}{3} \right)^x = 0 \][/tex]
Since function [tex]\( f(x) \)[/tex] does not converge to 0 based on the provided values, this statement is false.
4. Both functions are increasing on all intervals of [tex]\( x \)[/tex].
For [tex]\( f(x) \)[/tex], the table values show:
- [tex]\( f(0) = -12 \)[/tex] to [tex]\( f(1) = -4 \)[/tex] (increasing)
- [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = 0 \)[/tex] (increasing)
- [tex]\( f(2) = 0 \)[/tex] to [tex]\( f(3) = 2 \)[/tex] (increasing)
- [tex]\( f(3) = 2 \)[/tex] to [tex]\( f(4) = 3 \)[/tex] (increasing)
[tex]\( f(x) \)[/tex] is increasing on the given intervals.
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -12 \left( \frac{1}{3} \right)^x \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \left( \frac{1}{3} \right)^x \)[/tex] decreases, making [tex]\( g(x) \)[/tex] more negative, indicating that [tex]\( g(x) \)[/tex] is decreasing.
Thus, this statement is false.
5. Both functions approach [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
For [tex]\( f(x) \)[/tex]:
Based on the table, we don't have information for negative [tex]\( x \)[/tex] values, so it's inconclusive for [tex]\( f(x) \)[/tex].
For [tex]\( g(x) \)[/tex]:
As [tex]\( x \to -\infty \)[/tex],
[tex]\[ \left( \frac{1}{3} \right)^x \to \infty \][/tex]
[tex]\[ -12 \left( \frac{1}{3} \right)^x \to -\infty \][/tex]
Therefore, [tex]\( g(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Without information for [tex]\( f \)[/tex], we cannot conclusively state that this is true for both functions. Hence, this statement is also false.
### Conclusion
The true statement about the functions is:
- The functions have the same [tex]\( y \)[/tex]-intercept.
Thus, only the first statement is true.
### Function [tex]\( f(x) \)[/tex]
Function [tex]\( f \)[/tex] is defined by the values given in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & -12 & -4 & 0 & 2 & 3 \\ \hline \end{array} \][/tex]
### Function [tex]\( g(x) \)[/tex]
Function [tex]\( g \)[/tex] is given by the equation:
[tex]\[ g(x) = -12 \left( \frac{1}{3} \right)^x \][/tex]
### Analyzing Statements
1. The functions have the same [tex]\( y \)[/tex]-intercept.
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex].
For [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = -12 \][/tex]
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(0) = -12 \left( \frac{1}{3} \right)^0 = -12 \][/tex]
Both functions have [tex]\( y \)[/tex]-intercept at [tex]\( (0, -12) \)[/tex]. Therefore, this statement is true.
2. The functions have the same [tex]\( x \)[/tex]-intercept.
The [tex]\( x \)[/tex]-intercept occurs when the function value is 0.
For [tex]\( f(x) \)[/tex]:
From the table, [tex]\( f(2) = 0 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercept for [tex]\( f(x) \)[/tex] is [tex]\( x = 2 \)[/tex].
For [tex]\( g(x) \)[/tex]:
[tex]\[ -12 \left( \frac{1}{3} \right)^x = 0 \][/tex]
This equation does not have any real solution because [tex]\( -12 \left( \frac{1}{3} \right)^x \)[/tex] never equals 0 for any real [tex]\( x \)[/tex]. Therefore, [tex]\( g(x) \)[/tex] has no [tex]\( x \)[/tex]-intercept.
Hence, this statement is false.
3. Both functions approach the same value as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex].
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] for [tex]\( f(x) \)[/tex], the function values seem to increase slowly based on the table. For [tex]\( g(x) \)[/tex]:
[tex]\[ \lim_{x \to \infty} g(x) = \lim_{x \to \infty} -12 \left( \frac{1}{3} \right)^x = 0 \][/tex]
Since function [tex]\( f(x) \)[/tex] does not converge to 0 based on the provided values, this statement is false.
4. Both functions are increasing on all intervals of [tex]\( x \)[/tex].
For [tex]\( f(x) \)[/tex], the table values show:
- [tex]\( f(0) = -12 \)[/tex] to [tex]\( f(1) = -4 \)[/tex] (increasing)
- [tex]\( f(1) = -4 \)[/tex] to [tex]\( f(2) = 0 \)[/tex] (increasing)
- [tex]\( f(2) = 0 \)[/tex] to [tex]\( f(3) = 2 \)[/tex] (increasing)
- [tex]\( f(3) = 2 \)[/tex] to [tex]\( f(4) = 3 \)[/tex] (increasing)
[tex]\( f(x) \)[/tex] is increasing on the given intervals.
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = -12 \left( \frac{1}{3} \right)^x \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \left( \frac{1}{3} \right)^x \)[/tex] decreases, making [tex]\( g(x) \)[/tex] more negative, indicating that [tex]\( g(x) \)[/tex] is decreasing.
Thus, this statement is false.
5. Both functions approach [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
For [tex]\( f(x) \)[/tex]:
Based on the table, we don't have information for negative [tex]\( x \)[/tex] values, so it's inconclusive for [tex]\( f(x) \)[/tex].
For [tex]\( g(x) \)[/tex]:
As [tex]\( x \to -\infty \)[/tex],
[tex]\[ \left( \frac{1}{3} \right)^x \to \infty \][/tex]
[tex]\[ -12 \left( \frac{1}{3} \right)^x \to -\infty \][/tex]
Therefore, [tex]\( g(x) \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Without information for [tex]\( f \)[/tex], we cannot conclusively state that this is true for both functions. Hence, this statement is also false.
### Conclusion
The true statement about the functions is:
- The functions have the same [tex]\( y \)[/tex]-intercept.
Thus, only the first statement is true.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
