f(x) = 1/3x + 3 and g(x) = x - 2
The functions f and g above are defined for all values of x. In the xy-plane, what is the y-coordinate of the point of intersection of the graphs of f and g?
It is not clear if the x in the f(x) equation is in the denominator.
I assume you mean
f(x) = 1/(3x) + 3, and not
x/3 + 3.
f(x) = g(x) when
1/(3x) + 3 = x -2
1/(3x) = x - 5
3x^2 -15x -1 = 0
Solve for x. There will be two answers.
x = (1/6)[15 +/- sqrt(225+12)]
= 5.066 or -0.0658
f(x) = (1/3)x the fraction one third x
In that case, the problem simplifies to:
(x/3) + 3 = x -2
2x/3 = 5
x = 7.5
y = x-2 = 5.5
To find the y-coordinate of the point of intersection of the graphs of f and g, we need to find the x-value that makes f(x) equal to g(x). Once we have the x-coordinate, we can substitute it into either f(x) or g(x) to find the y-coordinate.
First, let's set f(x) equal to g(x):
1/3x + 3 = x - 2
To solve this equation, we need to get rid of the fractions. We can do so by multiplying both sides of the equation by the denominator, which is 3:
3 * (1/3x + 3) = 3 * (x - 2)
The 3 on the left side will cancel out the fraction:
1x + 9 = 3x - 6
Now, let's gather the x terms on one side of the equation and the constants on the other side:
1x - 3x = -6 - 9
Simplifying, we get:
-2x = -15
To isolate x, we divide both sides of the equation by -2:
x = -15 / -2
This simplifies to:
x = 15/2
Now that we have the x-coordinate of the point of intersection, we can substitute it into either f(x) or g(x) to find the y-coordinate. Let's substitute it into f(x):
f(x) = 1/3x + 3
f(15/2) = 1/3 * (15/2) + 3
To multiply fractions, multiply the numerators and multiply the denominators:
f(15/2) = 15/6 + 18/6
Adding the fractions:
f(15/2) = 33/6
Simplifying the fraction:
f(15/2) = 11/2
Therefore, the y-coordinate of the point of intersection of the graphs of f and g is 11/2.