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# Find analytical expression of an inverse function and a line function

Question:

An inverse function y = k/x and a line function y = mx + b intersects at point A (1, 2) and point B (n, -1). Find analytical expression of the inverse function and the line function.

Solution:

The blue curve is the inverse function y = k/x in which, x y are variables and k is an unknown number. The orange line is y = mx + b, in which m is the slope of the line and b is the y-intercept, both m and b are unknown numbers.

Because the point A (1, 2) lies on the blue curve, so, the point A (1, 2) satisfy the inverse function. Substitute the coordinate of the point A (1, 2) into the inverse function y = k/x, we get the value of k.

y = k/x
2 = k/1
so, k = 2

Thus, we get the inverse function y = 2/x. Now substitute the coordinate of the point B (n, -1) into the inverse function to get the unknow x-coordinate of the point B (n, -1)

y = 2/x
-1 = 2/n
so, n = -2

Thus, we get the coordinate of the point B (-2, -1). Now we use both point A (1, 2) and point B (-2, -1) to get the line function y = mx + b.

Substitute point A (1, 2) into the line function y = mx + b

y = mx + b
2 = m (1) + b
m + b = 2 ... name this as equation1

Substitute point B (-2, -1) into the line function y = mx + b

y = mx + b
-1 = m(-2) + b
-2m + b = -1 ... name this as equation2
equation1 - equation2
3m = 3
m = 1
substitute m = 1 into equation1
b = 2 - m = 2 - 1 = 1

Therefore, the line function is: y = x + 1 and the inverse function is y = 2/x. Watch the video for more details.