Find the linear to linear function f(x) satisfying the given requirements: f(10)=20, f(30)=25, and the graph of f(c) has y=30 as its horizontal asymptote
f(x) = (ax+b)/(x+d)
Since y=30 is the asymptote, a=30:
f(x) = (30x+b)/(x+d)
Now plug in your two points and solve for b,d.
To find the linear-to-linear function f(x) satisfying the given requirements, we can start by recognizing that a linear function has the form f(x) = mx + b, where m is the slope and b is the y-intercept.
We are given two points on the graph of f(x), namely (10, 20) and (30, 25). This allows us to find the slope of the function using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the given values, we have:
m = (25 - 20) / (30 - 10)
m = 5 / 20
m = 1/4
So, the slope of the linear function is 1/4.
Next, we need to determine the y-intercept (b). The function has a horizontal asymptote at y = 30. Since f(x) is a linear function, the y-value of the asymptote should be equal to the y-intercept.
Therefore, we set f(x) equal to y and solve for x:
30 = mx + b
Using the slope (m = 1/4), the equation becomes:
30 = (1/4)x + b
To find b, we can substitute one of the known points on the graph, such as (10, 20):
20 = (1/4)(10) + b
20 = 5/2 + b
b = 20 - 5/2
b = 15/2
So, the y-intercept (b) is 15/2.
Putting it all together, the linear-to-linear function f(x) satisfying the given requirements is:
f(x) = (1/4)x + 15/2