Problem 78 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$f(t)=\frac{5}{t^{2}-7 t+6}$$

Step-by-Step Solution

Answer

The domain of the function \(f(t)=\frac{5}{t^2-7t+6}\) is \(D = \{ t \in \mathbb{R} | t \neq 1, t \neq 6 \}\).

Step 1: Identify the denominator

The denominator of the function \(f(t)\) is given by the expression \((t^{2}-7 t+6)\). We need to find the values of \(t\) that make this expression equal to zero.

Step 2: Solve the quadratic equation in the denominator for t

We need to solve the following quadratic equation: \(t^{2}-7t+6=0\) We can try to factor this quadratic equation. If it cannot be factored, we could use the quadratic formula. But in this case, it can be factored as: \((t-6)(t-1)=0\) Now, we find the values of \(t\) that satisfy this equation: \(t-6=0\) or \(t-1=0\) Thus, \(t=6\) or \(t=1\)

Step 3: Define the domain by excluding the values that make the denominator zero

Since the function is undefined for \(t=6\) and \(t=1\), the domain will be all real values of \(t\) except these two values. We can write the domain as follows: \[ D = \{ t \in \mathbb{R} | t \neq 1, t \neq 6 \} \] Finally, the domain of the function \(f(t)=\frac{5}{t^2-7t+6}\) is \(D = \{ t \in \mathbb{R} | t \neq 1, t \neq 6 \}\).

Key Concepts

Rational Functions

Rational functions are a specific type of function quoted as a ratio of two polynomials. Just like a fraction, where you have a numerator and a denominator, a rational function takes the form \( \frac{P(t)}{Q(t)} \). Here, \( P(t) \) and \( Q(t) \) are polynomials, with \( Q(t) \) not equal to zero. This is crucial since dividing by zero results in an undefined value. In the exercise, \( f(t) = \frac{5}{t^2 - 7t + 6} \), the denominator \( t^2 - 7t + 6 \) diagnoses our function's domain. When working with rational functions, it is vital to ensure that the values assumed by the variables do not zero-out the denominator. This prevents running into undefined situations. Hence, understanding which values are not permitted is a key step in analyzing rational functions.

Quadratic Equations

Quadratic equations are a foundational element in algebra, represented in the form \( ax^2 + bx + c = 0 \). These equations have at most two solutions or roots. The form seen in the denominator of our function, \( t^2 - 7t + 6 = 0 \), is a quadratic equation. Solving it determines which values will make the function undefined due to the zero denominator.

• If it can be expressed as two distinct factors, setting each factor equal to zero will unveil the roots or solutions of the quadratic equation.
• If factoring is challenging, the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can always be a reliable fallback.

It is essential to solve it efficiently to find the critical values which, in our situation, are \( t = 1 \) and \( t = 6 \).

Factoring

Factoring is a mathematical strategy used to simplify expressions or resolve equations, especially quadratics. It involves expressing a polynomial as a product of its simpler polynomials. In this case, we factor \( t^2 - 7t + 6 \) as \( (t - 6)(t - 1) \). This process:

• Breaks down the quadratic equation into components that easily reveal solutions.
• Simplifies calculations by providing a straightforward method to solve equations efficiently.
• Ensures we find the values of \( t \) that make the denominator zero, exposing domain restrictions.

Aptly executing factoring requires insight into the multiplication and distribution of polynomials, ensuring precise solutions in less time.

Algebra

Algebra is often regarded as a mathematical language essential for solving equations and understanding a wide array of mathematical concepts and theories. It introduces tools and techniques necessary for simplifying and solving equations like the ones found in our exercise.

• It revolves around manipulating symbols and numbers to derive solutions or simplify expressions.
• Skills in algebra allow tackling complex mathematical problems, making it foundational for more advanced mathematics.


• In relation to our function \( f(t) \), algebra helps in solving the quadratic equation \( t^2 - 7t + 6 = 0 \) and provides methods such as factoring to find restrictions within the domain.

Algebra is crucial for students aiming to build competence in diverse mathematical areas, from basic problem-solving to more elaborate mathematical reasoning.