Answer :
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.
