Answer :
To analyze the functions [tex]\(g(x)\)[/tex] and [tex]\(t(x)\)[/tex], we need to examine their key characteristics: the [tex]\(y\)[/tex]-intercepts and the end behavior.
Step 1: Determining the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex]
To find the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex], we evaluate the function at [tex]\(x = 0\)[/tex]:
[tex]\[ g(x) = 9 \left(\frac{1}{3}\right)^x - 4 \][/tex]
[tex]\[ g(0) = 9 \left(\frac{1}{3}\right)^0 - 4 \][/tex]
[tex]\[ g(0) = 9 \cdot 1 - 4 \][/tex]
[tex]\[ g(0) = 9 - 4 \][/tex]
[tex]\[ g(0) = 5 \][/tex]
So, the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex] is [tex]\(5\)[/tex].
Step 2: Evaluating the end behavior of [tex]\(g(x)\)[/tex]
To understand the end behavior of [tex]\(g(x)\)[/tex], we look at the limits as [tex]\(x\)[/tex] approaches positive and negative infinity:
- As [tex]\(x \to \infty\)[/tex]:
[tex]\[ \lim_{x \to \infty} g(x) = \lim_{x \to \infty} \left[9 \left(\frac{1}{3}\right)^x - 4\right] = 9 \cdot 0 - 4 = -4 \][/tex]
- As [tex]\(x \to -\infty\)[/tex]:
[tex]\[ \lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} \left[9 \left(\frac{1}{3}\right)^x - 4\right] = 9 \cdot \infty - 4 = \infty \][/tex]
So, the end behavior of [tex]\(g(x)\)[/tex] is:
- [tex]\(g(x) \to -4\)[/tex] as [tex]\(x \to \infty\)[/tex]
- [tex]\(g(x) \to \infty\)[/tex] as [tex]\(x \to -\infty\)[/tex]
Step 3: Analyzing [tex]\(t(x)\)[/tex] (as visually represented)
The [tex]\(y\)[/tex]-intercept and end behavior of [tex]\(t(x)\)[/tex] are unspecified in the problem. We know the following from the information provided:
- The [tex]\(y\)[/tex]-intercept of [tex]\(t(x)\)[/tex] is unknown.
- The end behavior is also unspecified.
Given the precise information about the function [tex]\(g(x)\)[/tex] and the lack of information about [tex]\(t(x)\)[/tex], the correct comparison must note the uncertainty about [tex]\(t(x)\)[/tex].
Conclusion:
Given that [tex]\(t(x)\)[/tex] has unspecified [tex]\(y\)[/tex]-intercept and end behavior, we can only confidently state that:
The correct comparison statement between the two functions is:
- "They have different [tex]\(y\)[/tex]-intercepts and different end behavior," since the [tex]\(y\)[/tex]-intercept and end behavior of [tex]\(t\)[/tex] are both unknown and cannot be confirmed to match [tex]\(g(x)\)[/tex]'s values. Therefore, option C is the correct statement.
Step 1: Determining the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex]
To find the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex], we evaluate the function at [tex]\(x = 0\)[/tex]:
[tex]\[ g(x) = 9 \left(\frac{1}{3}\right)^x - 4 \][/tex]
[tex]\[ g(0) = 9 \left(\frac{1}{3}\right)^0 - 4 \][/tex]
[tex]\[ g(0) = 9 \cdot 1 - 4 \][/tex]
[tex]\[ g(0) = 9 - 4 \][/tex]
[tex]\[ g(0) = 5 \][/tex]
So, the [tex]\(y\)[/tex]-intercept of [tex]\(g(x)\)[/tex] is [tex]\(5\)[/tex].
Step 2: Evaluating the end behavior of [tex]\(g(x)\)[/tex]
To understand the end behavior of [tex]\(g(x)\)[/tex], we look at the limits as [tex]\(x\)[/tex] approaches positive and negative infinity:
- As [tex]\(x \to \infty\)[/tex]:
[tex]\[ \lim_{x \to \infty} g(x) = \lim_{x \to \infty} \left[9 \left(\frac{1}{3}\right)^x - 4\right] = 9 \cdot 0 - 4 = -4 \][/tex]
- As [tex]\(x \to -\infty\)[/tex]:
[tex]\[ \lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} \left[9 \left(\frac{1}{3}\right)^x - 4\right] = 9 \cdot \infty - 4 = \infty \][/tex]
So, the end behavior of [tex]\(g(x)\)[/tex] is:
- [tex]\(g(x) \to -4\)[/tex] as [tex]\(x \to \infty\)[/tex]
- [tex]\(g(x) \to \infty\)[/tex] as [tex]\(x \to -\infty\)[/tex]
Step 3: Analyzing [tex]\(t(x)\)[/tex] (as visually represented)
The [tex]\(y\)[/tex]-intercept and end behavior of [tex]\(t(x)\)[/tex] are unspecified in the problem. We know the following from the information provided:
- The [tex]\(y\)[/tex]-intercept of [tex]\(t(x)\)[/tex] is unknown.
- The end behavior is also unspecified.
Given the precise information about the function [tex]\(g(x)\)[/tex] and the lack of information about [tex]\(t(x)\)[/tex], the correct comparison must note the uncertainty about [tex]\(t(x)\)[/tex].
Conclusion:
Given that [tex]\(t(x)\)[/tex] has unspecified [tex]\(y\)[/tex]-intercept and end behavior, we can only confidently state that:
The correct comparison statement between the two functions is:
- "They have different [tex]\(y\)[/tex]-intercepts and different end behavior," since the [tex]\(y\)[/tex]-intercept and end behavior of [tex]\(t\)[/tex] are both unknown and cannot be confirmed to match [tex]\(g(x)\)[/tex]'s values. Therefore, option C is the correct statement.
