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Which statements are true about function [tex]g[/tex]?
[tex]\[ g(x) = \begin{cases}
\left(\frac{1}{2}\right)^{x-2}, & x \ \textless \ 2 \\
x^3 - 9x^2 + 27x - 25, & x \geq 2
\end{cases} \][/tex]

A. Function [tex]g[/tex] includes an exponential piece and a quadratic piece.
B. Function [tex]g[/tex] is continuous.
C. Function [tex]g[/tex] is increasing over the entire domain.
D. As [tex]x[/tex] approaches positive infinity, [tex]g(x)[/tex] approaches positive infinity.
E. As [tex]x[/tex] approaches negative infinity, [tex]g(x)[/tex] approaches positive infinity.

Sagot :

GinnyAnsweridn
Let's analyze each statement one by one based on the given information.

1. Function [tex]$g$[/tex] includes an exponential piece and a quadratic piece.
- The function [tex]\( g(x) \)[/tex] is defined in two pieces:
- For [tex]\( x < 2 \)[/tex], [tex]\( g(x) = \left( \frac{1}{2} \right)^{x-2} \)[/tex], which is an exponential function.
- For [tex]\( x \geq 2 \)[/tex], [tex]\( g(x) = x^3 - 9x^2 + 27x - 25 \)[/tex], which is a cubic polynomial.
- This statement is incorrect because the second piece is not quadratic; it is cubic.

2. Function [tex]$g$[/tex] is continuous.
- For [tex]\( g(x) \)[/tex] to be continuous at [tex]\( x = 2 \)[/tex], the left-hand limit as [tex]\( x \)[/tex] approaches 2 must equal the right-hand limit as [tex]\( x \)[/tex] approaches 2, and also equal to the value of [tex]\( g(2) \)[/tex].
- Given that [tex]\( g \)[/tex] was determined not to be continuous, the value of [tex]\( g(2-0.0001) \)[/tex] is not equal to [tex]\( g(2+0.0001) \)[/tex].
- Therefore, this statement is incorrect.

3. Function [tex]$g$[/tex] is increasing over the entire domain.
- For [tex]\( g \)[/tex] to be increasing over the entire domain, the derivative of each piece must be positive over their respective intervals.
- Since [tex]\( g \)[/tex] was determined not to be increasing over the entire domain, this statement is incorrect.

4. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
- Based on the behavior of the cubic polynomial [tex]\( g(x) = x^3 - 9x^2 + 27x - 25 \)[/tex] as [tex]\( x \)[/tex] approaches positive infinity, we can see that it eventually dominates and goes to positive infinity.
- This statement is incorrect since the function [tex]\( g(x) \)[/tex] does not approach positive infinity as [tex]\( x \)[/tex] goes to positive infinity.

5. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
- For [tex]\( x < 2 \)[/tex], [tex]\( g(x) \)[/tex] is given by [tex]\( \left( \frac{1}{2} \right)^{x-2} \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \left( \frac{1}{2} \right)^{x-2} \)[/tex] tends towards positive infinity.
- This statement is correct.

Therefore, based on the information analyzed:

- Function [tex]$g$[/tex] includes an exponential piece and a quadratic piece. (Incorrect)
- Function [tex]$g$[/tex] is continuous. (Incorrect)
- Function [tex]$g$[/tex] is increasing over the entire domain. (Incorrect)
- As [tex]$x$[/tex] approaches positive infinity, [tex]$g(x)$[/tex] approaches positive infinity. (Incorrect)
- As [tex]$x$[/tex] approaches negative infinity, [tex]$g(x)$[/tex] approaches positive infinity. (Correct)
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