Problem 66 - Calculus and its applications

Question

Is the function given by \(F(x)=\frac{1}{x^{2}-7 x+10}\) continuous at \(x=4 ?\) Why or why not?

Step-by-Step Solution

Answer

The function is continuous at \(x=4\). The denominator is not zero at this point.

Step 1: Identify Points of Discontinuity

A function can be discontinuous at a point if it contains a division by zero there. For the function \(F(x) = \frac{1}{x^2 - 7x + 10}\), the denominator becomes zero when \(x^2 - 7x + 10 = 0\). Let's find the roots of this quadratic equation to find points of potential discontinuity.

Step 2: Solve the Quadratic Equation

Solve the quadratic equation \(x^2 - 7x + 10 = 0\) using factorization. This factors into \((x-5)(x-2) = 0\). Solving \(x-5 = 0\) gives \(x = 5\), and solving \(x-2 = 0\) gives \(x = 2\). These are the points where the function may not be continuous.

Step 3: Check Continuity at x = 4

Since the points of discontinuity are \(x = 5\) and \(x = 2\), the function \(F(x)\) does not have division by zero at \(x = 4\). Therefore, the function is continuous at \(x = 4\) because it is defined and does not approach infinity or have other discontinuities at this point.

Key Concepts

Discontinuity points

In mathematics, continuity of a function is about how smooth its graph is, without any gaps, jumps, or sudden changes. Discontinuity points are those specific points where a function fails to be continuous. For rational functions like the one in our exercise, the main type of discontinuity occurs when the denominator equals zero. This is because dividing by zero is undefined.

To determine if and where these discontinuities occur, we set the denominator equal to zero and solve the resulting equation. If a specific value of the variable causes the denominator to become zero, then the function is discontinuous at that point. However, if the denominator is not zero at a particular point, it means the function is continuous there, unless there are other complications like vertical asymptotes or other types of singularities.

Understanding where these breakpoints occur is crucial for graphing the function and analyzing its behavior in different intervals of its domain.

Quadratic equation

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations frequently appear when dealing with functions like the denominator in our exercise, \( x^2 - 7x + 10 \). Solving quadratic equations is a key skill for finding discontinuity points in rational functions.

The most common methods to solve a quadratic equation include:

• Factorization: Express the equation as a product of two binomials and set each to zero.
• Quadratic formula: Use \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions.
• Completing the square: Rearrange the equation into a perfect square trinomial.

In our exercise, factorization was used to factor \( x^2 - 7x + 10 \) into \((x-5)(x-2) = 0\), revealing the roots \( x = 2 \) and \( x = 5 \). These roots indicate where the rational function may have discontinuities.

Rational functions

Rational functions are expressions formed by dividing two polynomials. They take the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). The function in this exercise, \( F(x) = \frac{1}{x^2 - 7x + 10} \), is a common example of a rational function.

Understanding rational functions requires analyzing their features, such as:

• Domain: The set of all possible input values (\( x \)) for which the function is defined. For rational functions, this excludes values that make the denominator zero.
• Asymptotes: Lines that the graph approaches but never touches. They can be vertical, like the points where the denominator is zero, or horizontal/oblique, depending on the degrees of the numerator and denominator polynomials.
• Discontinuity points: Values that cause the denominator to be zero, potentially leading to infinite values or undefined behavior.

By identifying the discontinuity points using the denominator's roots, we protect against undefined function evaluations and gain insights into the function's graphical behavior. In our case, knowing \( x = 2 \) and \( x = 5 \) are discontinuities helps predict and sketch the function's behavior around these values.