The graph of function [tex]\( t \)[/tex] is shown.

Function [tex]\( g \)[/tex] is represented by the equation:
[tex]\[ g(x) = 9\left(\frac{1}{3}\right)^2 - 4 \][/tex]

Which statement correctly compares the two functions?

A. They have the same [tex]\( y \)[/tex]-intercept and the same end behavior.
B. They have different [tex]\( y \)[/tex]-intercepts but the same end behavior.
C. They have different [tex]\( y \)[/tex]-intercepts and different end behaviors.



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.