What are the values of [tex]$x$[/tex] for which the denominator is equal to zero for [tex]$y=\frac{3 x-1}{x^2-4 x+4}$[/tex]?
Answer (2)
The problem asks for the values of x where the denominator of the given function is zero. We factor the denominator x 2 − 4 x + 4 into ( x − 2 ) ( x − 2 ) . Setting this equal to zero, we find that x = 2 . Therefore, the denominator is zero when x = 2 . x = 2
Explanation
Understanding the Problem We are given the function y = x 2 − 4 x + 4 3 x − 1 and asked to find the values of x for which the denominator is equal to zero. These values are also known as the points of discontinuity of the function.
Setting up the Equation To find the values of x that make the denominator zero, we need to solve the equation x 2 − 4 x + 4 = 0 . This is a quadratic equation, and we can solve it by factoring.
Factoring the Quadratic We can factor the quadratic expression as follows: x 2 − 4 x + 4 = ( x − 2 ) ( x − 2 ) = ( x − 2 ) 2
Solving for x Now we set the factored expression equal to zero and solve for x :
( x − 2 ) 2 = 0 Taking the square root of both sides, we get: x − 2 = 0 Adding 2 to both sides, we find: x = 2
Final Answer Thus, the denominator is equal to zero when x = 2 . This means that the function y has a point of discontinuity at x = 2 .
Examples
Understanding points of discontinuity is crucial in various real-world applications. For instance, in electrical engineering, the function describing the voltage in a circuit might have a discontinuity at a certain time, indicating a sudden change or a switch being flipped. Similarly, in physics, a function describing the density of a material might have a discontinuity at the boundary between two different materials. Identifying these points helps engineers and scientists understand and model the behavior of systems accurately.
The denominator of the function y = x 2 − 4 x + 4 3 x − 1 is equal to zero when x = 2 . Therefore, the function has a point of discontinuity at this value. The solution is that x = 2 .
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