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Question

Define the function f by

2x + 2 for x < 0

f(x) = ax + b for 0 < x < 3

4x - 5 for 3 < x

If a and b are chosen so as to make f(x) continuous everywhere, then a =

Answer 1

ax+ b must equal 2 when x=0 and 7 when x=3 in order for this to be continuous. From the first equation, b=2, so ax+2 = 7 when x=3 so...

a = 5/3

Answer 2

To be continuous,

ax+b = 4x-5 and
ax+b = 2x+2

ie a = a(x)

hence 2ax+2b = 6x-3
so a = 3x-1.5-b

Therefore by substitution,

3x^2 - (4.5+b)x + 5 - b = 0

This is a quadratic which may be solved to give b(x).

Then substitute b(x) to give

a(x)= 3x-1.5-b(x)

Answer 3

b=2
a=5/2