Problem 65 - Contemporary Precalculus

Question

$$\text {Find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=2 x^{2}-x+1 ; \quad g(x)=x^{2}-4 x+4$$

Step-by-Step Solution

Answer

Question: Find the values of x for which the following functions are equal: $$f(x) = 2x^2 - x + 1$$ $$g(x) = x^2 - 4x + 4$$ Answer: The values of x for which f(x) = g(x) are: $$x = \frac{-3 + \sqrt{21}}{2} \text{ and } x = \frac{-3 - \sqrt{21}}{2}$$

Step 1: Write down the given functions

We have the following functions: $$f(x) = 2x^2 - x + 1$$ $$g(x) = x^2 - 4x + 4$$

Step 2: Set f(x) equal to g(x)

We're looking for the values of x for which f(x) = g(x). To find them, set f(x) equal to g(x) and solve for x: $$2x^2 - x + 1 = x^2 - 4x + 4$$

Step 3: Simplify the equation

First, subtract the g(x) function from both sides of the equation to simplify it: $$2x^2 - x + 1 - (x^2 - 4x + 4) = 0$$ Now, distribute the negative sign to all terms in the parentheses and then combine like terms: $$x^2 + 3x - 3 = 0$$

Step 4: Solve for x

At this point, the equation is a quadratic equation, which can be solved using various methods such as factoring, the quadratic formula, or completing the square. In this case, we'll use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, a = 1, b = 3, and c = -3.

Step 5: Calculate x values using the quadratic formula

Plug the values of a, b, and c into the quadratic formula and solve for x: $$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-3)}}{2(1)}$$ $$x = \frac{-3 \pm \sqrt{21}}{2}$$ Thus, the values of x for which f(x) = g(x) are: $$x = \frac{-3 + \sqrt{21}}{2} \text{ and } x = \frac{-3 - \sqrt{21}}{2}$$

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression involving sums of powers of one variable with coefficients. In this case, both functions, \(f(x)\) and \(g(x)\), are quadratic polynomials because their highest degree is 2. Quadratic polynomials have the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).

For students, understanding the structure of polynomials is key to solving many algebraic problems. Each polynomial function can take different shapes known as parabolas when graphed on the Cartesian plane, and these can be useful for various applications in mathematics including curve fitting and modeling motion or growth patterns. In this exercise, we examined two quadratic functions to determine where their parabola graphs overlapped.

Root Finding

Root finding, sometimes known as solving equations, is the process of determining the values of \(x\) that make the equation true. In the context of this exercise, we were tasked with finding the values of \(x\) for which \(f(x) = g(x)\). To achieve this, we set their expressions equal and solved for \(x\).

This often involves rearranging and simplifying the equation until we reach a standard form, such as a quadratic equation, which you can then solve using reliable methods like factoring, the quadratic formula, or completing the square. Root finding is an essential concept in algebra, used extensively in solving real-world problems where one needs to find the intersection points of two functions.

Quadratic Formula

The quadratic formula is a universal tool we use to solve quadratic equations, typically formatted as \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For students, it's crucial to remember this formula as it not only finds solutions for quadratic equations irrespective of whether they can be factored but also gives insight into the discriminant \(b^2 - 4ac\). The discriminant helps understand the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, there's exactly one root (a repeated root).
• If negative, the equation has no real roots, only complex ones.

In our problem, using \(a = 1\), \(b = 3\), and \(c = -3\), the quadratic formula yielded two real solutions, which were the \(x\) values where \(f(x) = g(x)\).

Function Intersection

Function intersection involves finding the points where two or more functions meet or have the same output value. For functions \(f(x)\) and \(g(x)\), their intersection points are where their equations are equal. This is what leads to the concept of setting \(f(x) = g(x)\) and solving for \(x\).

In geometry, these points represent where the graphs of the functions cross on a coordinate plane. Intersections are vital in various fields of science and engineering, where they might help identify equilibrium points or solutions to systems of equations. In this exercise, the intersections of \(f(x)\) and \(g(x)\) give us insight into how the two quadratic functions relate to each other spatially.