Problem 17 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 34, the function is the set of all ordered pairs \((x, y)\) satisfying the given equation. Find the domain and range of the function, and draw a sketch of the graph of the function. $$ F: y=\frac{(x+1)\left(x^{2}+3 x-10\right)}{x^{2}+6 x+5} $$

Step-by-Step Solution

Answer

Domain: All real numbers except x = -5 and x = -1. Range: Real values of y influenced by behavior at x-excluded roles and function characteristics. Vertical asymptote at x = -5.

Step 1 - Identify the Domain

To find the domain of the function, determine all values of x for which the function is defined. The function is undefined where the denominator equals zero. Set the denominator equal to zero and solve for x: \[ x^2 + 6x + 5 = 0 \] Factor the quadratic equation: \[ (x+5)(x+1) = 0 \] The solutions are x = -5 and x = -1. Therefore, the domain consists of all real numbers except x = -5 and x = -1.

Step 2 - Determine the Range

To find the range, analyze the behavior of the function value y as x approaches critical points and asymptotes. Evaluate the function at different points and consider horizontal/vertical asymptotes and critical points. From the function: \[ y = \frac{(x+1)(x^2 + 3x - 10)}{x^2 + 6x + 5} \] The expression \[ (x+1) \] in the numerator and \[ (x+1) \] in the denominator cancel each other out leaving \[ y = \frac{(x^2 + 3x - 10)}{(x + 5)} \] This simplified function is easier to analyze for asymptotic behavior. As x approaches -5, the function goes to infinity, leading to vertical asymptote at x = -5. Thus, the range would cover all real values of y except where there are gaps due to vertical asymptotes, which would need to be more specifically analyzed based on the graph.

Step 3 - Draw the Graph

To draw the graph: 1. Plot the points where function is defined (excluding where x = -5 and x = -1). 2. Determine the vertical and horizontal asymptotes. Vertical asymptote at x = -5 as analyzed.3. Evaluate y-values near x = -5 and -1 to confirm behavior.4. Sketch the curve accordingly considering intercept at y when x equals 0, and other turning points derived from behavior analysis.

Key Concepts

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches. They are crucial in understanding the behavior of rational functions, especially as we approach certain critical values of x. There are two main types of asymptotes: vertical and horizontal. Vertical asymptotes occur where the denominator of a rational function equals zero because the function becomes undefined, leading to the value of y approaching infinity. For our given function, we identified vertical asymptotes at x = -5 and x = -1 by setting the denominator equal to zero and solving for x. Horizontal asymptotes, on the other hand, describe the behavior of the function as x approaches positive or negative infinity. For example, if the degrees of the highest power of x are the same in the numerator and denominator, the horizontal asymptote will be the ratio of their coefficients.

Graphing Rational Functions

Graphing rational functions involves several key steps to ensure accuracy. First, find the domain to determine where the function is defined. For the given function, the domain excludes x = -5 and x = -1 because the denominator equals zero at these points. Next, identify the asymptotes. For instance, vertical asymptotes at x = -5 indicate where the graph will rise or fall sharply without crossing those lines. Then, evaluate the function at various points to obtain y-values, especially near the asymptotes. Sketch the overall behavior by connecting these points smoothly, observing how the graph approaches the asymptotes. Always remember to check for horizontal or slant asymptotes as x approaches infinity. This will guide how the graph behaves in extreme values, ensuring you capture the full behavior of the function.

Factoring Quadratics

Factoring quadratics is a key step in simplifying and solving rational functions. Quadratic equations can often be factored into two binomials, making it easier to solve for zeros or simplify the function. For example, the denominator of our function, x^2 + 6x + 5, factors into (x + 5)(x + 1). This shows where the function is undefined and helps identify possible simplifying steps. In this case, since the numerator also has an (x + 1) term, they cancel each other out, simplifying the function further. Factoring not only aids in solving where the function equals zero (roots) but also helps in understanding the function's behavior. Always aim to factor quadratics whenever possible, as it simplifies the analysis and graphing of rational functions significantly.