Question

If \(y=\left[1 /\left(t^{2}+t-2\right)\right] y \operatorname{yk} t=[1 /(x-1)]\) then \(y\) is discontinuous at \(\mathrm{x} \in \ldots \ldots\) (a) \(\\{1,2\\}\) (b) \(\\{1,-2\\}\) (c) \(\\{1,(1 / 2), 2\\}\) (d) \(Z-\\{1,(1 / 2), 2\\}\)

Short answer

The function \(y\) is discontinuous at \(x \in \{1, -2\}\).

Step-by-step solution

Identify the denominator

In this problem, the denominator is the expression \(t^2 + t - 2\).

Find the values of t that make the denominator equal to zero

To do this, we need to solve the equation \(t^2 + t - 2 = 0\). We can apply quadratic formula or factor this expression: \(t^2 + t - 2 = (t + 2)(t - 1)\) Setting each factor to zero: \(t + 2 = 0 \Rightarrow t = -2\) \(t - 1 = 0 \Rightarrow t = 1\)

Answer the question

Now that we've found the values of \(t\) that make the denominator zero, we can see that the function \(y = \frac{1}{t^2 + t - 2} y\) is discontinuous at \(x = -2\) and \(x = 1\). Therefore, the correct answer is (b) \(\{1,-2\}\).

Key concepts

Quadratic equations

Quadratic equations are a central part of algebra and are used to describe parabolic curves. These equations typically take the form \( ax^2 + bx + c = 0 \), where \( a\), \( b\), and \( c \) are constants. The primary goal when dealing with quadratic equations is to find the values of \( x \) that make the equation true, known as the roots or solutions.

The solutions can be found using several techniques:

• Factoring: Expressing the quadratic as a product of two binomials.
• Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square: Rewriting the quadratic in perfect square form.

In some cases, the discriminant, \( b^2 - 4ac \), helps determine the number and nature of the roots. If the discriminant is positive, there are two real and distinct solutions. If zero, one real solution exists, and if negative, two complex solutions appear.

In the original exercise, identifying where the denominator equals zero involves solving a quadratic equation. This process ensures that we can find the points of discontinuity by determining where the function is undefined.

Factoring polynomials

Factoring polynomials is an important technique for simplifying expressions and solving equations. It revolves around dividing a polynomial into simpler "factor" polynomials that, when multiplied together, give back the original polynomial.

To factor polynomials like \( t^2 + t - 2 \), follow these steps:

• Identify two numbers that multiply to the constant term (in this case, \(-2\)) and add to the coefficient of the linear term (which is \(1\)).
• For the given expression, the numbers \(2\) and \(-1\) work since they satisfy both the sum and product conditions.
• Rewrite the middle term using these numbers: \( t^2 + 2t - t - 2 \).
• Group and factor by grouping: \((t(t + 2) - 1(t + 2))\).
• This results in \((t + 2)(t - 1)\), providing the factored form.

A successfully factored polynomial allows easier identification of the roots, showcasing where graphs intersect the x-axis or where expressions like our function become undefined.

Zeroes of a function

Zeroes of a function, also known as solutions or roots, are the values of \( x \) that make the function equal to zero. They are important in identifying key features of a graph and play an instrumental role in understanding the behavior of functions, particularly rational and polynomial functions.

When we find the zeroes of a quadratic function through factoring, as discussed before, we effectively identify the x-intercepts of the parabola. These zeroes are direct informative solutions that reveal significant points

• The graph of the function touches or crosses the x-axis at these points.
• They represent transition points, such as local maxima or minima, in real-world scenarios modeled by the function.

Alongside points at which a function is defined, zeroes dictate the domain and continuity, crucial for solving discontinuity problems like in the original exercise. By identifying the zeroes of the denominator, the exercise clarifies where the function ceases to be continuous, offering insights into its graphical behavior and defining its limits.