Question

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ h(x)=\frac{1}{x^{2}+2 x-15} $$

Short answer

The function \(h(x)=\frac{1}{x^{2}+2x-15}\) isn't continuous at \(x = 3\) and \(x = -5\).

Step-by-step solution

Set the Denominator Equal to Zero

Set the denominator of the function \(x^{2}+2x-15\) equal to zero. The equation becomes \(x^{2}+2x-15 = 0\).

Solve the Quadratic Equation

The equation \(x^{2}+2x-15 = 0\) is a quadratic equation and can be factored to \((x - 3)(x + 5) = 0\). Setting each bracketed term to zero gives the solutions \(x = 3\) and \(x = -5\).

Identify the Discontinuities

The function \(h(x)=\frac{1}{x^{2}+2x-15}\) is not continuous at \(x = 3\) and \(x = -5\), which are the solutions to the quadratic equation.

Key concepts

Graphing Functions

Graphing functions is a visual method that helps to understand the behavior of mathematical functions. When using a graphing utility, you're able to see the curve represented by the function and identify key features such as intercepts, turning points, and importantly, discontinuities.

To graph a function like the one given in the exercise, \(h(x)=\frac{1}{x^2+2x-15}\), you would typically enter it into a graphing calculator or software. As you vary the value of \(x\), the software computes the corresponding \(y\) values and plots the points on a coordinate plane. Connecting these points reveals the curve. Discontinuities, such as holes, jumps, or vertical asymptotes, become apparent - these are the \(x\)-values where the graph doesn't flow smoothly from one point to the next.

With the given function, as \(x\) approaches the values that make the denominator zero, the graph will show vertical asymptotes because the function's value heads towards infinity. This visual insight is pivotal in understanding where the function is not continuous.

Continuity in Functions

Continuity in functions is the property that describes if a function is unbroken or uninterrupted over its domain. For a function to be continuous at a certain point, three conditions must be met: the function must be defined at the point, the limit as it approaches the point must exist, and the function's value at that point must equal the limit.

In the given exercise, when we look at the function \(h(x)\) for continuity, there's an issue when the denominator becomes zero. At this moment, the function is not defined, hence violating the condition for continuity. By analyzing \(x^2+2x-15\), you can spot potential discontinuities by setting the denominator equal to zero, which essentially locates the \(x\)-values that make \(h(x)\) undefined.

The values of \(x = 3\) and \(x = -5\), found by solving the quadratic equation, show exactly where the function \(h(x)\) lacks continuity. These points correspond to vertical asymptotes on the graph where the function's value reaches infinity or negative infinity, hence the function 'breaks' and is not continuous at these points.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra. These are equations of the second degree, meaning they have the general form \(ax^2 + bx + c = 0\). To find values of \(x\) that satisfy this equation, various methods can be used, including factoring, completing the square, the quadratic formula, or graphing.

In the exercise, the quadratic equation \(x^2 + 2x - 15 = 0\) is factored into \( (x-3)(x+5) = 0\). This form reveals that \(x = 3\) and \(x = -5\) are the solutions since either factor set to zero solves the equation. These solutions are critical because they directly correlate with the discontinuities of the function \(h(x)\). It's important when graphing or analyzing the function's behavior to include these points since they represent limitations in the domain of \(h(x)\).

The ability to solve quadratic equations enables you to understand a variety of phenomena in mathematics, from the geometry of parabolas to the practical aspects of finding maximums and minimums in real-world applications.