* Consider the functions f(x)=1/x +4 and g(x)=2/x. Graph f and g on the same grid.

a)Determine the points of intersection of the two functions.

b)Show where f(x)<g(x).

c)Solve the equation 1/x +4=2/x to check your answer to part a).

d)Solve the inequality 1/x +4<2/x to check your answer to part b).

Here is a nice picture of your problem

http://www.wolframalpha.com/input/?i=plot+f%28x%29%3D1%2Fx+%2B4+%2C+g%28x%29%3D2%2Fx.+

disregard the "parametric graph"

Reiny i still kind of not understand this can you tell me the answers to each one please Reiny.. please....

To graph the functions f(x) = 1/x + 4 and g(x) = 2/x on the same grid, follow these steps:

Step 1: Determine the domain of the functions.
- Both f(x) and g(x) are defined for all x values except when the denominator is zero. Thus, x ≠ 0 for both functions.

Step 2: Choose some x-values and find the corresponding y-values for each function.
- Select several x-values, such as -3, -2, -1, 1, 2, 3, and calculate the corresponding y-values for each function.
- For f(x): Calculate f(-3), f(-2), f(-1), f(1), f(2), f(3).
- For g(x): Calculate g(-3), g(-2), g(-1), g(1), g(2), g(3).

Step 3: Plot the points on the graph.
- Use the x-values and their corresponding y-values to plot the points for each function on the graph.

Step 4: Connect the points to form the graphs.
- Connect the plotted points for f(x) and g(x) to obtain the graphs of the functions.

Now, let's move on to answering the specific questions:

a) Determine the points of intersection of the two functions.
- Find the x-values where f(x) = g(x).
- Solve the equation 1/x + 4 = 2/x to find the points of intersection.
- Multiply both sides of the equation by x to eliminate the denominators: x/x + 4x = 2.
- Simplify the equation: 1 + 4x = 2.
- Subtract 1 from both sides: 4x = 1.
- Divide both sides by 4: x = 1/4.
- Plug x = 1/4 into either f(x) or g(x) to find the corresponding y-value at the point of intersection.

b) Show where f(x) < g(x).
- To find where f(x) < g(x), set up the inequality f(x) < g(x).
- Substitute the function expressions: 1/x + 4 < 2/x.
- Multiply both sides by x to eliminate the denominators and ensure the inequality doesn't change direction.
- Simplify the inequality: 1 + 4x < 2.
- Subtract 1 from both sides: 4x < 1.
- Divide both sides by 4: x < 1/4.
- The solution to this inequality represents the region where f(x) < g(x) on the graph.

c) Solve the equation 1/x + 4 = 2/x to check your answer to part a).
- We already solved this equation in part a), which gave us x = 1/4 as the solution.
- You can substitute this solution back into the equation to verify that it holds true.

d) Solve the inequality 1/x + 4 < 2/x to check your answer to part b).
- We already solved this inequality in part b), which gave us x < 1/4 as the solution.
- You can choose an arbitrary value less than 1/4, substitute it into the inequality, and verify that it holds true.
- For example, substitute x = 0 into the inequality.
- Then, calculate 1/0 + 4 and 2/0 and check if 1/0 + 4 < 2/0 holds true.

These steps will help you graph the functions, find the points of intersection, and verify your answers to the equation and inequality.