Vladas believes that an equation with a squared term is never a function of x. Which equation can be used to show Vladas that his hypothesis is incorrect?
A. [tex]$x+y^2=25$[/tex]
B. [tex]$x^2-y=25$[/tex]
C. [tex]$x^2+y^2=25$[/tex]
D. [tex]$x^2-y^2=25[/tex]
Answer (2)
Analyze each equation to determine if it represents y as a function of x.
Solve each equation for y in terms of x.
Check if for every value of x, there is only one value of y.
Equation B, x 2 − y = 25 , is a function of x, thus showing Vladas's hypothesis is incorrect. B
Explanation
Analyzing the Problem Vladas believes that an equation with a squared term is never a function of x . We need to find an equation that demonstrates his hypothesis is incorrect. In other words, we need to find an equation with a squared term that is a function of x . A function of x means that for every value of x , there is only one corresponding value of y . We will analyze each option to see if it represents y as a function of x .
Checking Each Option Let's examine each option:
A) x + y 2 = 25 . Solving for y , we get y 2 = 25 − x , so y = ± 25 − x . For a single value of x , there are two possible values of y (a positive and a negative square root), so this is not a function of x .
B) x 2 − y = 25 . Solving for y , we get y = x 2 − 25 . For every value of x , there is only one value of y , so this is a function of x .
C) x 2 + y 2 = 25 . Solving for y , we get y 2 = 25 − x 2 , so y = ± 25 − x 2 . Again, for a single value of x , there are two possible values of y , so this is not a function of x .
D) x 2 − y 2 = 25 . Solving for y , we get y 2 = x 2 − 25 , so y = ± x 2 − 25 . As with options A and C, for a single value of x , there are two possible values of y , so this is not a function of x .
Conclusion Only option B, x 2 − y = 25 , represents y as a function of x . This equation includes a squared term ( x 2 ), but it is still a function of x . Therefore, this equation can be used to show Vladas that his hypothesis is incorrect.
Examples
In real life, functions are used to model relationships between different quantities. For example, the height of a ball thrown in the air can be modeled as a function of time. The equation y = x 2 − 25 could represent a simplified model of the height of an object ( y ) as it changes over time ( x ), where the object starts at a certain height and falls due to gravity. Understanding functions helps us predict and analyze these real-world phenomena.
The equation x 2 − y = 25 (Option B) is a function of x because it provides one specific value of y for each value of x. This demonstrates that Vladas's belief that equations with squared terms can never be functions is incorrect. Through analysis, we show that not all equations with squared terms fail to meet the definition of a function.
;
