Which equation is a function of [tex]$x$[/tex]?
[tex]$x=5$[/tex]
[tex]$x=y^2+9$[/tex]
[tex]$x^2=y$[/tex]
[tex]$x^2=y^2+16$[/tex]
Answer (1)
A function of x means that for every x value, there is only one corresponding y value.
Analyze each equation to see if it satisfies the condition.
x = 5 is not a function of x because x is constant and y can be any value.
x = y 2 + 9 and x 2 = y 2 + 16 are not functions of x because for one value of x , there are two values of y .
x 2 = y is a function of x because for every x value, there is only one y value: y = x 2 .
The equation that represents y as a function of x is x 2 = y .
Explanation
Analyzing the Problem We need to determine which of the given equations represents y as a function of x . In other words, for each value of x , there should be only one corresponding value of y . Let's analyze each equation.
Analyzing Equation 1
x = 5 : This equation represents a vertical line. For x = 5 , y can be any real number. Therefore, y is not a function of x .
Analyzing Equation 2
x = y 2 + 9 : To express y in terms of x , we can rearrange the equation as follows:
y 2 = x − 9
y = ± x − 9
For a single value of x , there are two possible values of y (a positive and a negative square root), so y is not a function of x .
Analyzing Equation 3
x 2 = y : This equation can be directly written as y = x 2 . For every value of x , there is only one value of y , which is x 2 . Therefore, y is a function of x .
Analyzing Equation 4
x 2 = y 2 + 16 : To express y in terms of x , we can rearrange the equation as follows:
y 2 = x 2 − 16
y = ± x 2 − 16
For a single value of x , there are two possible values of y (a positive and a negative square root), so y is not a function of x .
Conclusion Therefore, the equation that represents y as a function of x is x 2 = y .
Examples
In real life, functions are used to model relationships between different quantities. For example, the distance traveled by a car is a function of time, where for each moment in time, there is a unique distance the car has traveled. Similarly, the area of a circle is a function of its radius, where for each radius, there is a unique area. Understanding functions helps us predict and analyze these relationships.
