Problem 20 - 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. $$ h: y=\sqrt{6 x^{2}-5 x-4} $$

Step-by-Step Solution

Answer

Domain: (-\binfty, -0.5] \cup [ 4/3, \infty); Range: [0, \infty).

Step 1: Identify the constraint of the function

For the function to be defined, the expression inside the square root must be non-negative. This gives us the condition: i.e., $$6x^2-5x-4 \ge 0$$.

Step 2: Find the roots of the quadratic equation

To determine where the quadratic expression is zero, we find the roots of the equation: $$6x^2-5x-4 \= 0$$ Using the quadratic formula $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ where $$a=6$$, $$b=-5$$, $$c=-4$$, we get: $$x = \frac{5 \pm \sqrt{(-5)^2-4\cdot6\cdot(-4)}}{2\cdot6} \= \frac{5 \pm \sqrt{25+96}}{12} \= \frac{5 \pm11}{12}$$ which simplifies to: $$x = \frac{16}{12} \= \frac{4}{3}$$ and $$x = \frac{-6}{12} \= -0.5$$.

Step 3: Determine the intervals for the domain

Based on the roots, we can test the intervals around these roots to determine where the quadratic is non-negative: 1. When x < -0.5, $$6x^2-5x-4 > 0$$.2. When -0.5 \ \le x \le \ 4/3, $$6x^2 -5x -4 \ \lt 0$$.3. When x > \frac{4}{3}, $$6x^2-5x-4 > 0$$.

Step 4: Defining the domain

The function is defined where the quadratic expression is non-negative. Thus, the domain is: $$(-\binfty, -0.5] \cup [\ 4/3, \infty)$$.

Step 5: Range of the function

Since the range of the square root function is non-negative real numbers, the range is: $$[0, \infty)$$.

Step 6: Sketch the graph

The graph of the function can be sketched by identifying key points and understanding the shape of the curve. The function is defined for $$ x \ between (-\binfty, -0.5] \$$ and $$[4/3, \infty).$$ Plot the roots and shape the curve accordingly. Since we're dealing with a square root function, the graph will reflect the non-negative square root of the quadratic expression.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It is typically written in the form \(ax^2 + bx + c = 0\). To solve a quadratic equation, we can use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here, \(a\), \(b\), and \(c\) are constants where \(a eq 0\).
For instance, in our original exercise, we had to solve \(6x^2 - 5x - 4 = 0\).

Using the quadratic formula:
- Identify \(a = 6\)
- Identify \(b = -5\)
- Identify \(c = -4\)
By substituting these values into the formula, we found the roots to be \(x = \frac{4}{3}\) and \(x = -0.5\). These roots help us determine the intervals for the quadratic expression to be non-negative.

Understanding how to manipulate and solve quadratic equations is critical for finding the domain and range of functions involving these types of expressions.

Square Root Functions

Square root functions involve the square root of a variable or expression. They are typically written in the form \(y = \sqrt{f(x)}\), where \(f(x)\) must be non-negative because the square root of a negative number is not a real number.

In our example, we have \(y = \sqrt{6x^2 - 5x - 4}\). To ensure the function is defined, we need the expression inside the square root \((6x^2 - 5x - 4)\) to be \( \geq 0\).

By solving the quadratic equation, we identified the intervals where the expression under the square root is non-negative:
- For \(x < -0.5\), \(6x^2 - 5x - 4 > 0\)
- For \(-0.5 \leq x \leq \frac{4}{3}\), \(6x^2 - 5x - 4 \lt 0\)
- For \(x > \frac{4}{3}\), \(6x^2 - 5x - 4 > 0\)

Thus, the function \(y = \sqrt{6x^2 - 5x - 4}\) is defined for \((-\infty, -0.5] \cup [\frac{4}{3}, \infty)\).
Remember, the range of a square root function is always non-negative, so in this case, the range is \([0, \infty)\).

Graph Sketching

Sketching the graph of a function helps visualize its behavior and understand its constraints. For the function \(y = \sqrt{6x^2 - 5x - 4}\), we follow these steps:
1. **Identify key intervals and points**: Based on our domain, the function is defined for \(x\) values in the intervals \((-\infty, -0.5]\) and \([\frac{4}{3}, \infty)\).
2. **Plot the roots**: Mark the solutions to the quadratic equation \(x = -0.5\) and \(x = \frac{4}{3}\) on the x-axis.
3. **Shape of the curve**: Since we are dealing with a square root, the curve will reflect the positive (non-negative) values only.

Remember, the square root function's graph will be undefined where the quadratic expression is negative. Therefore, our graph will have breaks between the intervals. Connecting points in the specified intervals ensures an accurate representation.

Graph sketching is a crucial skill as it provides a visual interpretation of mathematical concepts, making it easier to grasp function behaviors and relationships.