Answer :
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)
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)
