1. 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)
Sorry but i forgot to pose what subject it was.
consider the graph of y = 1/x
I assume you know what it looks like.
f(x) = y+4, so it is the same curve shifted up by 4
g(x) = 2y, so it is the same curve scaled vertically by a factor of 2.
Visit
http://rechneronline.de/function-graphs/
and enter the functions. All will become clear. You might want to adjust the range to be -10 : 10, and plot a 3rd curve,
1/x+4 - 2/x
so you can see where f-g = 0.
To graph the functions f(x) = 1/(x) + 4 and g(x) = 2/(x) on the same grid, we can follow these steps:
Step 1: Determine the domain of each function. Since we have a fraction with x in the denominator, both functions will be undefined when x = 0. So the domain for both functions is all real numbers except x = 0.
Step 2: Find the intercepts:
- For f(x), when x = 1, f(1) = 1/1 + 4 = 5. So the point (1, 5) is on the graph of f(x).
- For g(x), when x = 1, g(1) = 2/1 = 2. So the point (1, 2) is on the graph of g(x).
Step 3: Choose additional points:
- For f(x), choose another point like x = -1. When x = -1, f(-1) = 1/(-1) + 4 = -3. So the point (-1, -3) is on the graph of f(x).
- For g(x), choose another point like x = -1. When x = -1, g(-1) = 2/(-1) = -2. So the point (-1, -2) is on the graph of g(x).
Step 4: Plot the points and draw smooth curves through them. Make sure to include arrows indicating the direction of the curves as they approach the asymptotes (x = 0).
a) To find the points of intersection of the two functions, we need to solve the equation f(x) = g(x):
1/(x) + 4 = 2/(x)
To find a common denominator, multiply both sides by x:
x + 4x = 2
Combine like terms:
5x = 2
Divide both sides by 5:
x = 2/5
So the two functions intersect at the point (2/5, f(2/5)) or (2/5, g(2/5)).
b) To determine where f(x) < g(x), we need to compare the values of f(x) and g(x) in various intervals:
- For x < 0:
- Choose x = -10: f(-10) = 1/(-10) + 4 = 3.9, g(-10) = 2/(-10) = -0.2. So f(x) > g(x) for x < 0.
- For 0 < x < 2/5:
- Choose x = 1: f(1) = 1/1 + 4 = 5, g(1) = 2/1 = 2. So f(x) > g(x) for 0 < x < 2/5.
- For x > 2/5:
- Choose x = 10: f(10) = 1/10 + 4 = 4.1, g(10) = 2/10 = 0.2. So f(x) > g(x) for x > 2/5.
Therefore, f(x) < g(x) for x < 0 and 0 < x < 2/5. On the graph, this can be shown by shading the region where the graph of f(x) is below the graph of g(x).
c) We can solve the equation 1/(x) + 4 = 2(x) to check our answer to part a):
1/(x) + 4 = 2x
Multiply both sides by x to clear the fraction:
1 + 4x = 2x^2
Rearrange to form a quadratic equation:
2x^2 - 4x - 1 = 0
Use the quadratic formula to solve for x:
x = (-(-4) ± √((-4)^2 - 4(2)(-1))) / (2(2))
= (4 ± √(16 + 8)) / 4
= (4 ± √24) / 4
= (4 ± 2√6) / 4
= (2 ± √6) / 2
So the solutions are x = (2 + √6) / 2 and x = (2 - √6) / 2. These two values confirm our answer to part a).
d) To check our answer to part b), we can solve the inequality 1/(x) + 4 < 2/(x):
1/(x) + 4 < 2/(x)
Multiply both sides by x:
1 + 4x < 2
Subtract 1 from both sides:
4x < 1
Divide both sides by 4:
x < 1/4
Therefore, the solution to the inequality is x < 1/4. This confirms our answer to part b) on the graph where f(x) < g(x).