Vladas believes that an equation with a squared term is never a function of [tex]$x$[/tex]. Which equation can be used to show Vladas that his hypothesis is incorrect?
[tex]$x+y^2=25$[/tex]
[tex]$x^2-y=25$[/tex]
[tex]$x^2+y^2=25$[/tex]
[tex]$x^2-y^2=25$[/tex]
Answer (2)
Analyze each equation to express y in terms of x .
Determine if each x value corresponds to a unique y value.
x 2 − y = 25 can be rewritten as y = x 2 − 25 , which is a function.
Therefore, the equation that disproves Vladas's hypothesis is x 2 − y = 25 .
Explanation
Understanding the Problem Vladas believes that any equation containing a squared term cannot represent a function of x . To disprove this, we need to identify an equation from the given options that includes a squared term but still defines y as a function of x . This means for every value of x , there should be only one corresponding value of y .
Analyzing Each Equation Let's examine each equation to see if we can express y as a function of x .
x + y 2 = 25 Solving for y , we get y 2 = 25 − x , which means y = ± 25 − x . For a single value of x , there are two possible values of y (one positive and one negative), so this is not a function of x .
x 2 − y = 25 Solving for y , we get y = x 2 − 25 . For each value of x , there is only one value of y , so this is a function of x .
x 2 + y 2 = 25 Solving for y , we get y 2 = 25 − x 2 , which means 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 .
x 2 − y 2 = 25 Solving for y , we get y 2 = x 2 − 25 , which means y = ± x 2 − 25 . Similar to the previous cases, for a single value of x , there are two possible values of y , so this is not a function of x .
Identifying the Correct Equation From the analysis above, the equation x 2 − y = 25 can be rewritten as y = x 2 − 25 . This equation represents a function because for every value of x , there is only one corresponding value of y . Therefore, this equation disproves Vladas's hypothesis.
Final Answer The equation x 2 − y = 25 can be used to show Vladas that his hypothesis is incorrect because it contains a squared term ( x 2 ) and is a function of x .
Examples
Imagine you're designing a parabolic mirror for a solar oven. The equation y = x 2 − 25 describes the shape of the mirror. For each point x along the horizontal axis, there is only one corresponding height y that defines the curve of the mirror. This ensures that sunlight is focused correctly onto a single point, maximizing the oven's efficiency. Understanding functions helps in designing precise and effective tools and technologies.
The equation x 2 − y = 25 shows that an equation with a squared term can still be a function of x since it gives a unique value of y for each x . All other options either yield two values for y or do not qualify as a function. Therefore, this particular equation disproves Vladas's hypothesis.
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