Find number a and b so that f(x) is continuous at every point

Question

f(x) = {1, x<0

{ax+b, 0 < x < 5

0, x>5

Answer 1

You have 2 unknowns: a and b

To be continuous at x = 0, ax + b must equal 1. thus: a*0 + b = 1, and therefore b = 1

To be continuous at x = 5, ax + b must equal 0. We already know b = 1, so: a*x + 1 = 0

But x = 5, so 5a + 1 = 0

Then:

5a = -1, and a = -1/5

Plot everything on a graph to check....

Answer 2

Find number a and b so that f(x) is continuous at every point

f(x) = {1, x<0

{ax+b, 0 < x < 5

0, x>5

For f(x) to be continuous at every point,it must be defined at x=0 as f(0) = 1, then it can change until f(5) = 0.

You may use a straight line (or other shape) to "connect these dots." When you have that segment in place, f(x) will be defined at any point x and it will be continuous.

The two-point form of the equation of a line is useful:

(y-y₁) = ( (y₂-y₁)/(x₂-x₁₎ ) (x - x₁) use points (0,1) and (5,0)

y - 1 = ( (-1/5) ( x - 0)

y = -x/5 + 1

This must match ax+b in the interval of 0 ≤ x ≤ 5.

a = -(1/5)

b = 1