Question

1. If a = 2 and b = 3, is f continuous at x = 1? Justify your answer.

2. Find a relationship between a and b for which f is continuous at x = 1.


3. Hint: A relationship between a and b just means an equation in a and b.

4. Find a relationship between a and b so that f is continuous at x = 2

Answer 1

For the function to be continous, the function has to be connected all through.
That means that this function must also be connected at x=1 and x=2


That means that for the function to be connected at x=1:

\(\color{black}{ \displaystyle "\left|x-1\right|+2" }\) evaluated at x=1,
must be equivalent to \(\color{black}{ \displaystyle "ax^2+bx" }\) evaluated at x=1.


And that as well means that for the function to be connected at x=2:

\(\color{black}{ \displaystyle "ax^2+bx" }\) evaluated at x=2,
must be equivalent to \(\color{black}{ \displaystyle "5x-10" }\) evaluated at x=2.

Answer 2

So therefore, must be that:

\(\color{black}{ \displaystyle |\color{red}{1}-1|+2=a\color{red}{(1)}^2+b\color{red}{(1)} }\)

\(\color{black}{ \displaystyle a\color{red}{(2)}^2+b\color{red}{(2)}=5\color{red}{(2)}-10 }\)

and this simplifies to the following system of equations:

\(\color{black}{ \displaystyle 2=a+b }\)

\(\color{black}{ \displaystyle 4a+2b=0 }\)