Problem 63 - Contemporary Precalculus

Question

$$\text {find the values of } x \text { for which } f(x)=g(x)$$. $$f(x)=2 x^{2}+13 x-14 ; \quad g(x)=8 x-2$$

Step-by-Step Solution

Answer

Answer: The functions f(x) and g(x) have the same output when x = 3/2 and x = -16/3.

Step 1: Set f(x) equal to g(x)

To find the values of x for which f(x) = g(x), set the two functions equal to each other: $$2x^2 + 13x - 14 = 8x - 2$$

Step 2: Rearrange the equation

Subtract 8x and add 2 to both sides of the equation to combine terms and create a quadratic equation: $$2x^2 + 13x - 8x - 14 - 2 = 0$$ This simplifies to: $$2x^2 + 5x - 16 = 0$$

Step 3: Factor the quadratic equation

Factor the quadratic equation to find its solutions: $$(2x - 3)(x + \frac{16}{3}) = 0$$

Step 4: Solve for x

Set each factor equal to zero and solve for x: $$2x - 3 = 0 \Rightarrow x = \frac{3}{2}$$ $$x + \frac{16}{3} = 0 \Rightarrow x = -\frac{16}{3}$$

Step 5: State the solution

The values of x for which f(x) = g(x) are \(\displaystyle x = \frac{3}{2}\) and \(\displaystyle x = -\frac{16}{3}\).

Key Concepts

Functions

Functions describe the relationship between input values and their corresponding outputs. In a function, each input value has one and only one output value. This makes understanding functions crucial for solving equations and understanding mathematical relationships.
In the given exercise, the two functions are defined as:

• Function \( f(x) = 2x^2 + 13x - 14 \) is a quadratic function. It involves an \( x^2 \) term, which suggests its graph is a parabola.
• Function \( g(x) = 8x - 2 \) is a linear function. It is characterized by a term in \( x \) with a straight-line graph.

To solve the exercise, you'll need to find where these functions intersect, meaning where their outputs (values of \( f(x) \) and \( g(x) \)) are equal for the same input \( x \).

Factoring

Factoring is the process of breaking down a complex expression into simpler expressions, which when multiplied together give the original expression.
To solve the quadratic equation \( 2x^2 + 5x - 16 = 0 \), factoring is a valuable method. It involves expressing the quadratic equation in the form of a product of two binomial expressions.
In the step-by-step solution, the equation \( 2x^2 + 5x - 16 = 0 \) is factored into: \((2x - 3)(x + \frac{16}{3}) = 0\)

• The first binomial, \(2x - 3\), corresponds to one solution for \( x \).
• The second, \(x + \frac{16}{3}\), represents the another solution for \( x \).

Factoring helps simplify equations to find solutions easily.

Solving Equations

Solving equations means finding the values of the variable that make the equation true. It's a core activity in algebra.
For the quadratic equation obtained from equating the two functions, \( 2x^2 + 5x - 16 = 0 \), we solve it by factoring, as demonstrated earlier. After factoring, each factor is set to zero, leading to simpler linear equations:

• \(2x - 3 = 0\). Solving this, we get \(x = \frac{3}{2}\).
• \(x + \frac{16}{3} = 0\). Solving this, we find \(x = -\frac{16}{3}\).

By solving, we determine the specific \( x \)-values where the original functions \( f(x) \) and \( g(x) \) are equivalent.

Graphical Intersection

Graphical intersection represents the points where the graphs of two functions meet. Understanding intersections graphically can enhance comprehension of the previous algebraic work.
A graph of \( f(x) = 2x^2 + 13x - 14 \) would typically be a U-shaped curve, given its quadratic nature. Meanwhile, \( g(x) = 8x - 2 \) would appear as a straight line with a constant slope.
For this exercise, the solutions \( x = \frac{3}{2} \) and \( x = -\frac{16}{3} \) correspond to the \( x \)-coordinates where these two graphs intersect. At these points:

• The output (or \( y \)-value) of both functions is exactly the same.
• This confirms the solutions found algebraically.

Thus, graphical intersection provides a visual proof and deeper understanding of where the two functions are equal.