Which statements about the function [tex]f[/tex] are true? Check all that apply.
The domain of [tex]f(x)[/tex] is all real numbers.
The range of [tex]f(x)[/tex] is all real numbers.
The domain of [tex]f(x)[/tex] is [tex]{x \mid x>0}[/tex].
The range of [tex]f(x)[/tex] is [tex]{y \mid y>0}[/tex].
[tex]f(2)=4[/tex]
[tex]f(2)=2(\frac{1}{2} x+3)[/tex]
Answer (2)
Evaluate f ( 2 ) using the given expression: f ( 2 ) = 2 ( 2 1 ( 2 ) + 3 ) = 8 .
Check if f ( 2 ) = 4 is true: Since f ( 2 ) = 8 , the statement f ( 2 ) = 4 is false.
Determine if the statements about the domain and range are true: We cannot determine the domain and range of f ( x ) with the given information.
The statement f ( 2 ) = 2 ( 2 1 x + 3 ) is true for x = 2 .
Explanation
Analyzing the Statements We are given several statements about a function f ( x ) and need to determine which are true. Let's analyze each statement.
Evaluating f(2) First, let's evaluate the expression for f ( 2 ) given by the last statement: f ( 2 ) = 2 ( 2 1 x + 3 ) . Substituting x = 2 , we get f ( 2 ) = 2 ( 2 1 ( 2 ) + 3 ) = 2 ( 1 + 3 ) = 2 ( 4 ) = 8 .
Checking f(2) = 4 Now, let's check the statement f ( 2 ) = 4 . Since we found that f ( 2 ) = 8 , this statement is false.
Analyzing Domain and Range Since we only have an expression for f ( 2 ) , we cannot determine the domain and range of the function f ( x ) . Therefore, we cannot determine if the statements about the domain and range are true or false.
Examples
Understanding functions and their properties, such as domain, range, and specific values, is crucial in many real-world applications. For example, in physics, the trajectory of a projectile can be modeled as a function of time. Knowing the domain and range of this function helps determine the valid time intervals and the possible heights the projectile can reach. Similarly, in economics, cost functions and revenue functions are used to model business operations. Understanding their domains and ranges helps in making informed decisions about production levels and pricing strategies.
We cannot confirm statements about the domain and range of f ( x ) without additional information. The statement f ( 2 ) = 4 is false, while the expression given for f ( 2 ) is true. Hence, the only statement we can verify as true is that f ( 2 ) = 2 ( 2 1 x + 3 ) .
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