Problem 63
Question
Find the domain of the function. Write your answer in set-builder notation. $$ g(t)=\frac{5-t}{t^{2}-t-2} $$
Step-by-Step Solution
Verified Answer
\( g(t) \)'s domain is \( \{ t \in \mathbb{R} \mid t \neq 2 \text{ and } t \neq -1 \} \).
1Step 1: Analyze the Denominator
Identify the denominator of the function, which is \( t^2 - t - 2 \). To find the domain, we need the denominator to not equal zero, since division by zero is undefined.
2Step 2: Set the Denominator Equal to Zero
Solve for \( t \) where the denominator is equal to zero: \( t^2 - t - 2 = 0 \).
3Step 3: Factor the Quadratic Equation
The quadratic \( t^2 - t - 2 \) can be factored as \( (t - 2)(t + 1) = 0 \). Identify the values of \( t \) that make the product zero.
4Step 4: Solve for Critical Values
From \( (t - 2)(t + 1) = 0 \), we find the solutions \( t = 2 \) and \( t = -1 \). These are the values that make the denominator zero and must be excluded from the domain.
5Step 5: Identify the Domain
The domain of \( g(t) \) includes all real numbers except the values \( t = 2 \) and \( t = -1 \). Express this in set-builder notation: \( \{ t \in \mathbb{R} \mid t eq 2 \text{ and } t eq -1 \} \).
Key Concepts
Set-builder notationQuadratic equationDivision by zero
Set-builder notation
When working with functions, especially in mathematics, it's important to specify the domain clearly. The domain consists of all possible input values (usually represented by variable(s)) for which the function is defined.
Set-builder notation is a concise way of describing a set by specifying the properties its members must satisfy. It uses the notation: \( \{ x \mid \text{condition} \} \), which reads as "the set of all \( x \) such that the condition holds".
For example, when determining the domain of a function that is not defined for certain values, set-builder notation is especially useful. As outlined in the given problem where \( g(t) = \frac{5 - t}{t^2 - t - 2} \), the denominator \( t^2 - t - 2 \) cannot be zero, as this would make the function undefined.
Through solving the quadratic equation related to the denominator, the critical values \( t = 2 \) and \( t = -1 \) were identified. Therefore, the domain can be expressed using set-builder notation as: \( \{ t \in \mathbb{R} \mid t eq 2 \text{ and } t eq -1 \} \). This describes all real numbers excluding \( t = 2 \) and \( t = -1 \), ensuring a clear understanding of where the function is defined.
Set-builder notation is a concise way of describing a set by specifying the properties its members must satisfy. It uses the notation: \( \{ x \mid \text{condition} \} \), which reads as "the set of all \( x \) such that the condition holds".
For example, when determining the domain of a function that is not defined for certain values, set-builder notation is especially useful. As outlined in the given problem where \( g(t) = \frac{5 - t}{t^2 - t - 2} \), the denominator \( t^2 - t - 2 \) cannot be zero, as this would make the function undefined.
Through solving the quadratic equation related to the denominator, the critical values \( t = 2 \) and \( t = -1 \) were identified. Therefore, the domain can be expressed using set-builder notation as: \( \{ t \in \mathbb{R} \mid t eq 2 \text{ and } t eq -1 \} \). This describes all real numbers excluding \( t = 2 \) and \( t = -1 \), ensuring a clear understanding of where the function is defined.
Quadratic equation
A quadratic equation is any equation that can be rearranged in standard form to \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). Quadratic equations are fundamental in algebra and appear frequently in various mathematical problems.
In the provided example, the denominator \( t^2 - t - 2 \) can lead to a quadratic equation of \( t^2 - t - 2 = 0 \). Solving this will help identify any values that make the denominator zero, which are the values to exclude from the function's domain.
One effective method of solving quadratic equations is factoring. You aim to express the quadratic in the form \( (t - r)(t - s) = 0 \), where \( r \) and \( s \) are the roots. By observing how the equation was factored in the exercise: \( (t - 2)(t + 1) = 0 \), we find the roots are \( t = 2 \) and \( t = -1 \).
In the provided example, the denominator \( t^2 - t - 2 \) can lead to a quadratic equation of \( t^2 - t - 2 = 0 \). Solving this will help identify any values that make the denominator zero, which are the values to exclude from the function's domain.
One effective method of solving quadratic equations is factoring. You aim to express the quadratic in the form \( (t - r)(t - s) = 0 \), where \( r \) and \( s \) are the roots. By observing how the equation was factored in the exercise: \( (t - 2)(t + 1) = 0 \), we find the roots are \( t = 2 \) and \( t = -1 \).
- These solutions are precisely the values that make the original expression equal zero.
- Knowing how to factor is crucial as it allows you to determine critical points quickly.
Division by zero
Division by zero is a mathematical concept best described as undefined. In mathematics, any expression that requires dividing by zero lacks meaning, since no number exists that multiplied by zero will yield a non-zero numerator.
In many mathematical functions, particularly rational functions where a variable appears in the denominator, division by zero is a potential problem. When a division by zero occurs, the function does not have a defined value for that particular input.
In our given function \( g(t) = \frac{5 - t}{t^2 - t - 2} \), division by zero must be avoided to find the valid domain. This necessitates identifying the values of \( t \) that result in a zero in the denominator. Solving the equation \( t^2 - t - 2 = 0 \) and finding \( t = 2 \) and \( t = -1 \), highlights precisely where the 'disallowed values' come from.
In many mathematical functions, particularly rational functions where a variable appears in the denominator, division by zero is a potential problem. When a division by zero occurs, the function does not have a defined value for that particular input.
In our given function \( g(t) = \frac{5 - t}{t^2 - t - 2} \), division by zero must be avoided to find the valid domain. This necessitates identifying the values of \( t \) that result in a zero in the denominator. Solving the equation \( t^2 - t - 2 = 0 \) and finding \( t = 2 \) and \( t = -1 \), highlights precisely where the 'disallowed values' come from.
- These values are excluded from the domain since they make the expression undefined.
- Understanding this concept helps prevent mistakes in calculations and ensures clarity when defining mathematical solutions.
Other exercises in this chapter
Problem 63
The volume of a cylinder is given by \(V=\pi r^{2} h,\) where \(r\) is the radius and \(h\) is the height. If the height of a cylindrical can is 6 inches and th
View solution Problem 63
Solve. Write answers in standard form. $$ x(x-4)=-5 $$
View solution Problem 64
A rectangle is 4 feet longer than it is wide. If the area of the rectangle must be less than or equal to 672 square feet, find the possible values for the width
View solution Problem 64
Use transformations to sketch a graph of \(f\). \(f(x)=2+\sqrt{-(x-3)}\)
View solution