Select all the correct answers.
Which statements are true about function [tex]$f$[/tex]?
[tex]$f(x)=\left\{\begin{array}{ll}\left(\frac{1}{3}\right)^x, & x \leq 1 \\-x^2+2 x-1, & x \textgreater 1\end{array}\right.$[/tex]
A. Function [tex]$f$[/tex] is continuous.
B. Function [tex]$f$[/tex] is decreasing over the entire domain.
C. As [tex]$x$[/tex] approaches positive infinity, [tex]$f(x)$[/tex] approaches positive infinity.
D. The domain of function [tex]$f$[/tex] is all real numbers.
E. The value of [tex]$f(1)$[/tex] is 2.
Answer (2)
Check continuity at x = 1 : lim x → 1 − f ( x ) = 3 1 and lim x → 1 + f ( x ) = 0 , so f is not continuous.
Analyze if f is decreasing: For x ≤ 1 , f ( x ) = ( 3 1 ) x is decreasing. For 1"> x > 1 , f ( x ) = − ( x − 1 ) 2 is decreasing. Thus, f is decreasing over the entire domain.
Evaluate the limit as x → ∞ : lim x → ∞ f ( x ) = − ∞ , so f ( x ) does not approach positive infinity.
Determine the domain: f is defined for all real numbers.
Evaluate f ( 1 ) : f ( 1 ) = 3 1 = 2 .
Therefore, the correct statements are that the function is decreasing over the entire domain and the domain of the function is all real numbers. So the final answer is:
Function f is decreasing over the entire domain, The domain of function f is all real numbers.
Explanation
Analyzing the Function We are given a piecewise function:
1 \end{array}\right."> f ( x ) = { ( 3 1 ) x , − x 2 + 2 x − 1 , x ≤ 1 x > 1
We need to determine which of the following statements are true:
Function f is continuous.
Function f is decreasing over the entire domain.
As x approaches positive infinity, f ( x ) approaches positive infinity.
The domain of function f is all real numbers.
The value of f ( 1 ) is 2 .
Analyzing Each Statement Let's analyze each statement:
Continuity:
For x ≤ 1 , f ( x ) = ( 3 1 ) x . At x = 1 , f ( 1 ) = ( 3 1 ) 1 = 3 1 .
For 1"> x > 1 , f ( x ) = − x 2 + 2 x − 1 . As x approaches 1 from the right, f ( x ) approaches − ( 1 ) 2 + 2 ( 1 ) − 1 = − 1 + 2 − 1 = 0 .
Since the left-hand limit ( x → 1 − ) is 3 1 and the right-hand limit ( x → 1 + ) is 0 , the function is not continuous at x = 1 .
Therefore, the statement 'Function f is continuous' is false .
Decreasing:
For x ≤ 1 , f ( x ) = ( 3 1 ) x is a decreasing exponential function.
For 1"> x > 1 , f ( x ) = − x 2 + 2 x − 1 = − ( x − 1 ) 2 . This is a downward-opening parabola with a vertex at x = 1 . For 1"> x > 1 , as x increases, f ( x ) decreases.
Since f ( 1 ) = 3 1 and lim x → 1 + f ( x ) = 0 , the function is decreasing at x = 1 as well.
Therefore, the statement 'Function f is decreasing over the entire domain' is true .
Limit as x approaches infinity:
As x approaches positive infinity, we consider the part of the function defined for 1"> x > 1 , which is f ( x ) = − x 2 + 2 x − 1 = − ( x − 1 ) 2 .
As x → ∞ , − ( x − 1 ) 2 → − ∞ .
Therefore, the statement 'As x approaches positive infinity, f ( x ) approaches positive infinity' is false .
Domain:
The function is defined for x ≤ 1 and 1"> x > 1 . Together, these cover all real numbers.
Therefore, the statement 'The domain of function f is all real numbers' is true .
Value at f(1):
f ( 1 ) = ( 3 1 ) 1 = 3 1 .
Therefore, the statement 'The value of f ( 1 ) is 2' is false .
Conclusion Based on the analysis, the true statements are:
Function f is decreasing over the entire domain.
The domain of function f is all real numbers.
Examples
Imagine you're designing a temperature control system where the temperature changes based on different conditions. The function f(x) could represent the temperature at time x. Understanding continuity helps ensure smooth temperature transitions, while knowing if the temperature is always decreasing is crucial for preventing overheating. Determining the domain tells you for what times the system is defined, and the function's value at a specific time gives you the exact temperature at that moment. Analyzing such functions is vital for designing reliable and safe control systems.
The true statements about the function f are that it is decreasing over the entire domain and its domain is all real numbers. It is not continuous, does not approach positive infinity as x goes to infinity, and the value of f ( 1 ) is 3 1 , not 2.
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