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Chapter 7: Problem 48

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=-\frac{3}{5 x}, g(x)=-x$$

Short Answer

Expert verified
The functions \(f(x)\) and \(g(x)\) intersect at \(x=-\frac{3}{5}\).

Step by step solution

01

Graphing the Functions

With the help of a graphing calculator, graph the two functions, \(f(x)=-\frac{3}{5x}\) and \(g(x)=-x\), on the same set of axes. Both functions are defined for all x except x=0.
02

Identifying the intersection points

Observe the graph and note down the points where the graphs of \(f(x)\) and \(g(x)\) intersect.
03

Solving for intersection points

To find the exact intersection points, solve the equation where \(f(x)=g(x)\), that is \(-\frac{3}{5x}=-x\). This implies \(x=-\frac{3}{5}\). Hence, the graphs intersect at \(x=-\frac{3}{5}\).

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Key Concepts

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

Intersection Points
Intersection points are where two or more graphs meet, showing the exact values that satisfy multiple equations at the same time. For the given functions,
  • the intersection point is where the values of both functions are equal when plotted on the same graph.
  • These points are crucial as they signify a shared solution between the equations.
To find these, you can look at the graph to visually identify them. Sometimes the calculations are straightforward and you can solve them algebraically too. In this exercise, the intersection point is at \(-\frac{3}{5}\).Graphical intersection:
  • Use a graphing tool to make this process smoother.
  • Visually verify where the graphs meet on the coordinate plane.
Understanding intersection points helps us grasp the relationship between two functions, particularly how and where they equate each other.
Graphing Calculator
Graphing calculators are versatile tools for visualizing complex functions to find their graphical intersections. These devices have functions such as:
  • plotting multiple graphs simultaneously, allowing us to see where they intersect.
  • zooming in and out of graphs for more detailed analysis on specific regions.
  • providing quick visuals of the behavior of functions, making learning intuitive.
For the given functions, a graphing calculator plots both \(f(x)=-\frac{3}{5x}\) and \(g(x)=-x\) on a single graph. This makes it easy to visually spot intersection points and see how each function behaves in comparison to the other.A graphing calculator is preferred for quick insights into function behaviors and identification of potential intersection points.
Equation Solving
Solving equations involves finding the values of the variable that make two expressions equal. For our exercise, this means determining when:\[-f(x)=-\frac{3}{5x} = g(x)=-x\]The steps for solving this algebraically include:
  • Set the two expressions equal to each other: \[-\frac{3}{5x}=-x\].
  • Simplify this by multiplying or dividing as necessary to solve for \(x\).
  • In this case, it simplifies to finding \(x = -\frac{3}{5}\).
Equation solving translates the visual and graphical data from graphing calculators into exact values, providing analytical accuracy. It's a compelling way to validate what you see on a graph with precise numeric solutions.

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Most popular questions from this chapter

Your friend claims that the least common multiple of two numbers is always greater than each of the numbers. Is your friend correct? Justify your answer.

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{15}{x} $$

Rewrite the function \(g\) in the form \(g(x)=\frac{a}{x-h}+k\). Graph the function. Describe the graph of \(g\) as a transformation of the graph of \(f(x)=\frac{a}{x}\). \(g(x)=\frac{2 x+3}{x}\)

Is \(f(x)=\frac{-3 x+5}{2^x+1}\) a rational function? Explain your reasoning.

Simplify the complex fraction. \(\frac{\frac{1}{3 x^2-3}}{\frac{5}{x+1}-\frac{x+4}{x^2-3 x-4}}\)
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