Question

Suppose f(x)={5x7x+c for x≥9, for x<9. In order to make f(x) continuous for all x in (−[infinity],[infinity]), what number should c be equal to? c= 9 (If there is no possible value of c that makes f(x) continuous for all real x, enter "none".)

Answer 1

`c=-18`

Explanation:

We have been given a piece-wise function. We are asked to find the value of possible value of c that will make the function continuous for all x in
`(-\infty,\infty)`.

`\left \{ {{f(x)=5x,\text{ for }x\geq 9} \atop {f(x)=7x+c,\text{ for }x>9}} \right.`

We know that a piece-wise function is continuous when right hand side limit is equal to left hand side limit.

To find the value of c that will make function continuous, we need to find the value of c at
`x=9` by equating both side functions as:

`7x+c=5x`

`7(9)+c=5(9)`

`63+c=45`

`63-63+c=45-63`

`c=-18`

Therefore,
`c=-18` will make the function continuous for all x in
`(-\infty,\infty)`.