Problem 28 - Calculus Early Transcendentals

Question

An important question about many functions concerns the existence and location of fixed points. A fixed point of \(f\) is a value of \(x\) that satisfies the equation \(f(x)=x ;\) it corresponds to a point at which the graph off intersects the line \(y=x\). Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=5-x^{2}$$

Step-by-Step Solution

Answer

Question: Find the fixed points of the function \(f(x) = 5-x^2\). Answer: The fixed points of the function are \(x = \frac{-1 + \sqrt{21}}{2}\) and \(x = \frac{-1 - \sqrt{21}}{2}\).

Step 1: Write the function

The function given in the problem is: $$f(x) = 5-x^2$$

Step 2: Set up the equation

To find the fixed points of the function, we need to solve the equation \(f(x) = x\). Substitute the expression for \(f(x)\) in the equation: $$5-x^2 = x$$

Step 3: Rearrange the equation

Rearrange the equation to form a quadratic equation: $$x^2 + x - 5 = 0$$

Step 4: Solve the quadratic equation

Now solve the quadratic equation to find the fixed points. This equation can be solved using factoring, completing the square, or the quadratic formula. In this case, the quadratic formula will be used: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where \(a=1\), \(b=1\), and \(c=-5\). Substitute the values of \(a\), \(b\), and \(c\) in the formula: $$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-5)}}{2(1)}$$

Step 5: Calculate the solutions

Simplify the expression to find the fixed points: $$x = \frac{-1 \pm \sqrt{21}}{2}$$ So, the fixed points are \(x = \frac{-1 + \sqrt{21}}{2}\) and \(x = \frac{-1 - \sqrt{21}}{2}\) .

Key Concepts

Quadratic Equation

A quadratic equation is one of the most fundamental structures in algebra. At its core, a quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The equation we derived, \(x^2 + x - 5 = 0\), is a perfect example.

Quadratics have two solutions, which can be found through various methods:

• Factoring: Splitting the middle term and finding what values of \(x\) make the equation zero.
• Completing the Square: Rewriting the equation in the form \((x+p)^2 = q\).
• Quadratic Formula: Using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which works universally for solving any quadratic equation.

For our exercise, we used the quadratic formula given complexity introduced by the negative constant, and discovered two distinct solutions, \(x = \frac{-1 \pm \sqrt{21}}{2}\). Each method relies on understanding the role of the coefficients and how they influence the graph and solutions of the equation.

The solutions represent the x-values where the function intersects the x-axis. Specific methods like factoring might not always work, especially with non-integer roots. Therefore, the quadratic formula is a reliable method ensuring we find the correct roots, even if they are irrational.

Graphing Functions

Visualizing a function is a powerful tool in understanding its behavior, and this especially applies to finding fixed points. A graph gives you a picture of where the function meets the line \(y = x\).

For the function \(f(x) = 5 - x^2\), graphing involves a few key steps:

• Identify the vertex: This function is a downward-facing parabola because of the \(-x^2\) term. The vertex is the highest point due to this orientation.
• Plotting points: Pinpointing where \(f(x)\) might intersect \(y = x\) helps in arriving at initial guesses for fixed points. Plotting several points can help visualize the curve's path and breadth.
• Finding intersections: The fixed points intersection \(f(x) = x\) visually represent where the function overlaps with the identity line \(y = x\). These are the solutions from our quadratic equation!

Graphing enhances comprehension by allowing students to visually confirm their algebraic steps. In this exercise, our solutions \(x = \frac{-1 + \sqrt{21}}{2}\) and \(x = \frac{-1 - \sqrt{21}}{2}\) would appear where the parabola and the line \(y = x\) intersect on this graph. Seeing the function and line interact simplifies the process of understanding how algebraic solutions correspond to geometric interpretations.

Solving Equations

Solving equations is a foundational skill in mathematics. When tasked with finding fixed points, solving the equation efficiently is critical.

In our task, we effectively turned the problem of finding fixed points into solving a quadratic equation \(x^2 + x - 5 = 0\). Here are some fundamental points used in solving equations like these:

• Setting up the equation: Establish a logical connection between what needs to be solved and the algebraic representation. In fixed points, it was \(f(x) = x\).
• Manipulate the structure: Rearrange terms to achieve a common form like the quadratic structure. Knowing the target form complicates solving only slightly due to familiarity with formulaic methods.
• Implementing the quadratic formula: Sometimes the chosen method is based on ease and accuracy. The quadratic formula is a systematic approach that applies broadly and yields both real and complex roots.
• Checking solutions: Once solutions are found, substituting back into the original setup ensures they're correct. This step ensures understanding and validation of your solutions' correctness.

Equation-solving skills not only help in mathematics but aid in developing logical reasoning and problem-solving abilities across various disciplines. For quadratic equations, like the one we solved, pattern recognition, and method adaptability are key.