Which statements about the function $f$ are true? Check all that apply.
The domain of $f(x)$ is {all real numbers}.
The range of $f(x)$ is {all real numbers}.
The domain of $f(x)$ is ${x \mid x>0}$.
The range of $f(x)$ is ${y \mid y>0}$.
$f(2)=4$
$f(2)=2(\frac{1}{2} x+3)$
Answer (2)
The domain and range statements cannot be evaluated without knowing the function f ( x ) .
Evaluate f ( 2 ) using the expression 2 ( 2 1 x + 3 ) at x = 2 , which gives f ( 2 ) = 8 .
Compare the calculated value f ( 2 ) = 8 with the statement f ( 2 ) = 4 , showing that it is false.
Conclude that none of the statements can be determined to be true. Therefore, the answer is that none of the statements are true.
Explanation
Analyzing the Statements We are given several statements about a function f and asked to determine which are true. Let's analyze each statement.
Domain and Range The statements about the domain and range of f ( x ) are:
The domain of f ( x ) is {all real numbers}.
The range of f ( x ) is {all real numbers}.
The domain of f ( x ) is { 0"> x ∣ x > 0 }.
The range of f ( x ) is { 0"> y ∣ y > 0 }. Without knowing the explicit form of the function f ( x ) , we cannot determine its domain and range. Therefore, we cannot definitively say whether these statements are true or false.
Evaluating f(2) The remaining statements are:
f ( 2 ) = 4
f ( 2 ) = 2 ( 2 1 x + 3 ) Statement 6 provides a definition for f ( 2 ) . We can evaluate this expression at x = 2 to find the value of f ( 2 ) .
Calculating f(2) Substituting x = 2 into the expression 2 ( 2 1 x + 3 ) , we get: f ( 2 ) = 2 ( 2 1 ( 2 ) + 3 ) = 2 ( 1 + 3 ) = 2 ( 4 ) = 8 So, f ( 2 ) = 8 .
Comparing the Results Comparing this result with statement 5, which says f ( 2 ) = 4 , we see that statement 5 is false since we found that f ( 2 ) = 8 . Statement 6 gives an expression for f ( 2 ) which evaluates to 8. However, statement 6 is not a valid statement as it contains the variable x when we are evaluating f ( 2 ) . Therefore, statement 6 is also false.
Conclusion Therefore, none of the statements can be determined to be true with the given information.
Examples
Understanding the domain and range of functions is crucial in many real-world applications. For example, when modeling the height of a projectile over time, the domain is restricted to non-negative time values since time cannot be negative. Similarly, the range might be limited by the maximum height the projectile can reach. Analyzing functions helps us make sense of constraints and possibilities in various scenarios, from physics to economics.
None of the statements about the function f can be verified as true. The evaluation shows that f ( 2 ) = 8 , making the statements f ( 2 ) = 4 and the proposed expression false. Therefore, none of the provided statements are correct.
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