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Chapter 3: Problem 63

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

Short Answer

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

Step by step solution

01

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: $$2x^2 + 13x - 14 = 8x - 2$$
02

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: $$2x^2 + 5x - 16 = 0$$
03

Factor the quadratic equation

Factor the quadratic equation to find its solutions: $$(2x - 3)(x + \frac{16}{3}) = 0$$
04

Solve for x

Set each factor equal to zero and solve for x: $$2x - 3 = 0 \Rightarrow x = \frac{3}{2}$$ $$x + \frac{16}{3} = 0 \Rightarrow x = -\frac{16}{3}$$
05

State the solution

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Example \(11(b)\) showed how we create a table of values for a function when you get to choose all the values of the inputs. The technique presented does not work for Casio calculators. This exercise is designed for users of Casio calculators. \(\cdot \quad\) Enter an equation such as \(y=x^{3}-2 x+3\) in the equation memory. This can be done by selecting TABLE in the MAIN menu. \- Return to the MAIN menu and select LIST. Enter the numbers at which you want to evaluate the function as List 1 \(\cdot \quad\) Return to the MAIN menu and select TABLE. Then press SET-UP [that is, 2nd MENU] and select LIST as the Variable; on the LIST menu, choose List 1. Press EXIT and then press TABL to produce the table. \- Use the up/down arrow key to scroll through the table. If you change an entry in the X column, the corresponding \(y_{1}\) value will automatically change. (a) Use this technique to duplicate the table in Example \(11(\mathrm{b})\) (b) Change the number -11 to \(10,\) and confirm that you've obtained \(10^{3}-2(10)+3\)

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=\sqrt{x^{3}+5} ; \quad g(x)=-\frac{1}{2} \sqrt{x^{3}+5}-6$$

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=x^{2}+x ; \quad g(x)=(x-3)^{2}+(x-3)+2$$

Let \(f(x)=x^{2}+5,\) and let \(g(x)=f(x-1)\) (a) Write the rule of \(g(x)\) and simplify. (b) Find the difference quotients of \(f(x)\) and \(g(x)\) (c) Let \(d(x)\) denote the difference quotient of \(f(x) .\) Show that the difference quotient of \(g(x)\) is \(d(x-1)\)

Compute and simplify the difference quotient of the function. $$f(x)=x^{2}+3 x-1$$
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