(1 point) Find the value of the constant c that makes the following function continuous on (-∞,∞).

Answer:

8/3

Step-by-step explanation:

To make this function be continuous, we need to make the one-sided limits of the function at x=3 equal.

this means that 3c+8=9c-8

8=6c-8

16=6c

c=16/6 = 8/3

The value of constant c that makes the given function continuous at (-∞ , ∞) is 6.

What do you mean by continuity of a function ?

Continuity of a function meant all the points in the given interval satisfy the function.

We know that the conditions for which a function is continuous at b = x is :

a) f(x) exists

b) [tex]\lim_{ b\to x}[/tex] or f(b) exists.

c) [tex]\lim_{b \to x}[/tex] f(b) = f(x)

We know that for any value of c , for b ∈ ( - ∞ , 3] the function is continuous and for b ∈ ( 3 , ∞) the function is continuous. So , for any value of c, the first two conditions are satisfied.

Now we need to check for the continuity of the function at b = 3.

or

[tex]\lim_{b \to 2}[/tex] f(b) = c × 3 + 8

or

[tex]\lim_{b \to 2}[/tex] f(b) = 3c + 8

Now , check for the continuity of the function for other interval at b = 3.

[tex]\lim_{b \to 2}[/tex] f(b) = c × 9 - 8

or

[tex]\lim_{b \to 2}[/tex] f(b) = 9c - 8

To be continuous both values should be equal.

i.e.,

3c + 8 = 9c - 8

The value of constant c will be :

6c = 18

or

c = 6

Therefore , the value of constant c that makes the given function continuous at (-∞ , ∞) is 6.

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